A simulation method for vibration control response of a rotating MFC laminate

Abstract

The application discloses a simulation method for vibration control response of a rotating MFC laminated plate, performs unit discretization on a flexible plate and an MFC plate, adopts an absolute node coordinate method to describe the motion and deformation of the flexible plate and the MFC plate, obtains a coordinate conversion matrix between the flexible plate unit and the upper MFC plate unit in combination with a laminated plate unit modeling method, calculates the generalized elastic force and the stiffness matrix of the flexible plate unit from the elastic potential energy of the flexible plate unit, introduces a constitutive equation of the MFC to describe the force-electricity coupling characteristics of the MFC, determines the generalized elastic force matrix, the generalized stiffness matrix and the generalized piezoelectric force matrix of the MFC plate unit, obtains the control voltage parameters according to a PD control strategy, determines the unit mass matrix, the stiffness matrix and the external force matrix of the MFC laminated plate unit, assembles the MFC laminated plate unit, obtains the dynamic equation of the rotating MFC laminated plate, and obtains the deformation and the velocity of the rotating MFC laminated plate based on the absolute node coordinate method. α Method solves the absolute node coordinate of the MFC laminated plate, and obtains the deformation and the velocity of the rotating MFC laminated plate.

Description

Technical Field

[0001] This invention relates to multibody system dynamics modeling technology, specifically to a simulation method for vibration control response of a rotating MFC laminate. Background Technology

[0002] Rotating components such as helicopter rotors and solar panels are widely used in the aerospace field. These flexible structures are prone to vibration problems caused by external excitation when undergoing large-scale rotational motion, which can affect their operation or lead to instability. Therefore, research on their dynamics and vibration control is particularly important. Macro fiber composites (MFCs), as a novel piezoelectric smart material, have advantages such as good flexibility and high strain-driven slope. Using them as actuators can effectively control the vibration response of structures. Therefore, establishing the dynamic equations of rotating MFC laminates and studying their vibration control response is of great significance.

[0003] In his paper "Mechanical Model of Macro Fiber Composite Structure and Active Control of Vibration in Plate and Shell Structures," Luo Wei equates the driving effect of MFC on the structure to external forces and torques, and studies the vibration control model of MFC composite structures. After reducing the order of the model, he completes simulation analysis and active vibration control experiments on flat plates and curved plates. However, the object he studies is a cantilever plate, and the mechanical properties of the laminated plate will be different under the influence of centrifugal force when it rotates. In his paper "Utilizing Macro Fiber Composite to Control Rotating Blade Vibrations," Hamed uses MFC to mitigate the nonlinear vibration phenomenon of rotating blades. The results show that before control, the blades experienced severe vibration and abrupt jumping behavior due to the presence of bifurcation points. After control, the blades exhibit a stable motion state due to the disappearance of bifurcation points. However, his description of the electromechanical coupling characteristics of MFC is not sufficient or accurate enough. Summary of the Invention

[0004] This invention proposes a simulation method for vibration control response of rotating MFC laminates.

[0005] The technical solution of this invention is: a simulation method for vibration control response of a rotating MFC laminate, comprising the following steps:

[0006] Step 1: Establish a physical model of the rotating MFC laminate, which consists of a flexible plate and an MFC plate, with the MFC plate located on the upper side of the flexible plate. Set the geometric parameters, material parameters, and motion parameters of the flexible plate, as well as the position parameters of the MFC plate.

[0007] Step 2: Discretize the flexible plate and the MFC plate into elements, use the absolute node coordinate method to describe the motion and deformation of the flexible plate and the MFC plate, and combine the laminated plate element modeling method to obtain the coordinate transformation matrix between the flexible plate element and the upper MFC plate element.

[0008] Step 3: Calculate the generalized elastic force and stiffness matrix of the flexible plate unit from the elastic potential energy of the flexible plate unit, introduce the constitutive equation of MFC to describe its electromechanical coupling characteristics, and use it as the actuator in the rotating MFC laminate to determine the generalized elastic force matrix, generalized stiffness matrix and generalized piezoelectric force matrix of the MFC plate unit, and obtain the control voltage parameters according to the PD control strategy.

[0009] Step 4: Obtain the element mass matrix, stiffness matrix and external force matrix of the MFC laminate element through coordinate transformation matrix, assemble the MFC laminate element, and obtain the dynamic equation of rotating MFC laminate by combining the constraint equation.

[0010] Step 5: Solve the dynamic equation of the rotating MFC laminate based on the generalized-α method, calculate the absolute node coordinates of the MFC laminate, and obtain the deformation and velocity of the rotating MFC laminate.

[0011] A simulation system for the vibration control response of a rotating MFC laminate is provided to implement the simulation method for the vibration control response of the rotating MFC laminate and to realize the simulation of the vibration control response of the rotating MFC laminate.

[0012] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the simulation method for vibration control response of a rotating MFC laminate, thereby simulating the vibration control response of the rotating MFC laminate.

[0013] A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, the simulation method for vibration control response of a rotating MFC laminate is implemented, thereby simulating the vibration control response of the rotating MFC laminate.

[0014] Compared with the prior art, the present invention has the following significant advantages: (1) The absolute node coordinate method is used to discretize the flexible plate, which can more accurately describe the deformation and motion of the plate element. The established equation has a constant mass matrix and there is no Coriolis force or centrifugal force. (2) The modeling method of the laminated plate element is a layered model. While maintaining the modeling accuracy of the laminated plate element with sufficient coupling conditions, it reduces the number of node coordinates of the laminated plate element by half, improves the calculation efficiency, and can directly describe the force-electric coupling characteristics of MFC instead of equating it to a single force. Attached Figure Description

[0015] Figure 1 This is a flowchart of the present invention.

[0016] Figure 2 This is a schematic diagram of a rotating MFC laminate.

[0017] Figure 3 Schematic diagram of flexible plate unit

[0018] Figure 4 This is a schematic diagram of a laminate unit.

[0019] Figure 5 This is the initial interface of the app.

[0020] Figure 6 This is the interface after inputting parameters.

[0021] Figure 7 The graph shows the vibration response over time at a rotational speed of 100 rad / min. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0023] This invention provides a simulation method for vibration control response of a rotating MFC laminate, comprising the following steps:

[0024] Step 1: Establish the physical model of the rotating MFC laminate;

[0025] MFC laminate consists of a flexible board and an MFC board, with the MFC board located on the upper side of the flexible board.

[0026] (1) Parameters of flexible plates

[0027] Geometric parameters: length L, width W, and thickness t.

[0028] Material parameters: density ρ A Elastic modulus E and Poisson's ratio v.

[0029] Motion parameters: Rotational speed w.

[0030] (2) Parameters of MFC board

[0031] Position parameters.

[0032] Step 2: Discretize the flexible board and the MFC board to obtain the flexible board unit and the MFC board unit. Combine the laminated board unit modeling method to obtain the coordinate transformation matrix between the flexible board unit and the upper MFC board unit.

[0033] The flexible plate and the MFC plate are discretized into m elements in the length direction and n elements in the width direction, resulting in a finite number of flexible plate elements and MFC plate elements.

[0034] like Figure 2 As shown, a global coordinate system is established with the centroid of the left end face of the MFC laminate as the origin, the length direction as the X-axis, the width direction inward as the Y-axis, and the thickness direction upward as the Z-axis. Figure 4 As shown, a coordinate system for the flexible plate unit is established with the midpoint of the vertical edge on the outer left side of the flexible plate unit as the origin, the X-axis along the length direction, the Y-axis along the width direction inward, and the Z-axis along the thickness direction upward.

[0035] The position r of any point P on the flexible plate element can be represented in the global coordinate system as:

[0036]

[0037] Where r x r y r z These are the coordinates of point P on the coordinate axes.

[0038] The position of point P can be represented by a shape function and absolute nodal coordinates.

[0039] r = [S1I S2I…S 16 I]e A =S A e A (2) Where e A The coordinates of the absolute element nodes of the flexible plate include 3 position coordinates and 9 partial derivatives of the position coordinates with respect to x, y, and z. It is a 3×3 identity matrix. S A The shape function of the flexible plate element is expressed as:

[0040]

[0041] In the formula, x, y, and z are the coordinates of point P in the coordinate system of the flexible plate element, respectively, and a, b, and t are the length, width, and thickness of the flexible plate element, respectively.

[0042] MFC laminate element modeling requires the following two basic assumptions to be met:

[0043] 1) Before and after the deformation of the MFC laminate unit, the tangential planes of the upper and lower unit end faces are always coplanar;

[0044] 2) During the movement and deformation of the flexible plate unit and the MFC plate unit, the contact surface never slips relative to each other.

[0045] Based on these two fundamental assumptions, such as Figure 2 As shown, the laminate consists of board A and board B, with board B glued directly above board A. A M is the midpoint of the line connecting the two nodes on the left end of board A. B E is the midpoint of the line connecting the two nodes on the left end of board B. A For M A With M B The intersection of the line connecting the two sides and the upper left boundary of plate A, E B For M A With M B The intersection of the line connecting the two sides and the lower left boundary of plate B, M A M B E A E B The four points are always collinear, E A E B The two points always coincide. Therefore, E can be obtained. A E B The absolute nodal coordinates of the two points are equal:

[0046]

[0047] In the formula, r EA r EB E respectively A E B The positions of two points in the global coordinate system E respectively A E B The partial derivatives of the positions of two points in the global coordinate system with respect to x in their respective unit coordinate systems. E respectively A E B The partial derivatives of the positions of two points in the global coordinate system with respect to y in their respective unit coordinate systems. E respectively A E B The partial derivatives of the positions of two points in the global coordinate system with respect to z in their respective unit coordinate systems.

[0048] E A With E B The absolute nodal coordinates of two points can be expressed using the shape functions and absolute nodal coordinates of the upper and lower plates, respectively:

[0049]

[0050]

[0051] In the formula, S A S B These are the shape function matrices for flexible boards and MFC boards, respectively. Let x be the partial derivatives of the shape function matrices of the flexible board and the MFC board, respectively. Let be the partial derivatives of the shape function matrices of the flexible board and the MFC board with respect to y, respectively. Let z be the partial derivatives of the shape function matrices of the flexible plate and the MFC plate, respectively, with respect to z. E respectively A Point and E B absolute node coordinates of point e A and e B The partial derivative, e A and e B These are the absolute node coordinates of the flexible board element and the MFC board element, respectively. A t represents the thickness of the flexible plate unit. B This represents the thickness of the MFC board unit.

[0052] Substituting formulas (5) and (6) into (4) yields:

[0053]

[0054] Similarly, for the other six points F A F B G A G B H A H B 、(F A The intersection of the line connecting the two nodes on the right side of plate A and the line connecting the two nodes on the right side of plate B with the upper boundary of the right side of plate A is F. B The intersection of the line connecting the two nodes on the right side of plate A and the line connecting the two nodes on the right side of plate B, and the lower boundary of the right side of plate B; G A The intersection of the line connecting the two nodes at the front end of plate A and the line connecting the two nodes at the front end of plate B with the upper boundary of the front end of plate A is G. B The intersection of the line connecting the two nodes at the front end of plate A and the line connecting the two nodes at the front end of plate B, and the lower boundary of the front end of plate B; H A The intersection of the line connecting the two nodes at the rear end of plate A and the line connecting the two nodes at the rear end of plate B with the upper boundary of the rear end of plate A is H. B For a line connecting two points: the midpoint of the line connecting the two nodes at the rear end of plate A and the midpoint of the line connecting the two nodes at the rear end of plate B, and the intersection point with the lower boundary of the rear end of plate B; similarly, we have:

[0055]

[0056] F respectively A Point and F B absolute node coordinates of point e A and e B The partial derivative, G A Point and G B absolute node coordinates of point e A and e B The partial derivative, H respectively A Point and H B absolute node coordinates of point e A and e B The partial derivative of .

[0057] Combining formulas (7) and (8), we can obtain:

[0058]

[0059] In the formula, T BA This is the transformation matrix of absolute node coordinates between the upper and lower plate units.

[0060] Step 3: Derive the generalized elastic force and stiffness matrix of the flexible plate element from its elastic potential energy. Introduce the constitutive equation of MFC to describe its electromechanical coupling characteristics, and use it as the actuator in the rotating MFC laminate to derive the generalized elastic force matrix, generalized stiffness matrix, and generalized piezoelectric force matrix of the MFC plate element. The specific method is as follows:

[0061] Elastic potential energy E of flexible plate unit ε for:

[0062]

[0063] In the formula, σ A ε A V A c A These are the stress, strain, volume, and elastic coefficient matrices of a flexible plate element, respectively. A The expression is:

[0064]

[0065] In the formula, E is the elastic modulus of the flexible plate, and ν is the Poisson's ratio of the flexible plate.

[0066] Taking the partial derivative of equation (10) with respect to the nodal coordinates of the flexible plate element, we can obtain the generalized elastic force of the flexible plate element as follows:

[0067]

[0068] In the formula K A This is the stiffness matrix of the flexible plate element.

[0069] The generalized mass matrix of the flexible plate element is:

[0070]

[0071] As the actuator in the MFC laminate, the MFC plate exhibits electromechanical coupling characteristics. Under the action of an external electric field, it will undergo structural deformation. Introducing the constitutive equation of the MFC plate into the flexible plate element model, its constitutive relation can be expressed as:

[0072]

[0073] In the formula, σ B For the stress matrix of the MFC plate element, c B Let ε be the elastic coefficient matrix of the MFC plate element. B For the strain matrix of the MFC plate element, e E Let E be the piezoelectric coupling coefficient matrix of the MFC board element, D be the electric displacement of the MFC board element, ζ be the relative permittivity matrix of the MFC board element, and E be the piezoelectric coupling coefficient matrix of the MFC board element. e The magnitude of the electric field intensity applied to the MFC plate unit is:

[0074]

[0075] In the formula, U is the voltage applied to the MFC board unit, and h e This refers to the electrode spacing in the MFC board unit.

[0076] The required control voltage is obtained based on the PD control strategy. The PD control method is as follows:

[0077]

[0078] In the formula, k p k is the proportional gain coefficient. d Let be the differential time constant, and let error(t) be the displacement in the Z direction at the free end of the MFC laminate. The velocity in the Z direction of the free end of the MFC laminate.

[0079] From the constitutive equation of the MFC plate element, the strain energy of the MFC plate element can be obtained as follows:

[0080]

[0081] In the formula V B This represents the volume of an MFC board unit.

[0082] Taking the partial derivative of the strain energy equation with respect to the absolute nodal coordinates of the MFC plate element yields the generalized elastic force of the MFC plate element:

[0083]

[0084] The stiffness matrix of an MFC plate element can be obtained by taking the partial derivative of the generalized elastic force of the MFC plate element with respect to the nodal coordinates of the MFC plate element:

[0085]

[0086] The work done by the electric field force acting on the MFC board unit is:

[0087]

[0088] The generalized electric force can be obtained by taking the partial derivative of the work done by the electric field force with respect to the nodal coordinates of the MFC plate element:

[0089]

[0090] In the formula K MFCE This is the stiffness matrix of the electric field force of the MFC plate element.

[0091] The generalized mass matrix of the MFC board element is

[0092]

[0093] Step 4: Obtain the element mass matrix, stiffness matrix, and external force matrix of the MFC laminate element through coordinate transformation matrix, that is, map the stiffness matrix, elastic force matrix, and external force matrix of the MFC plate element to the flexible plate element. Then, assemble the MFC laminate element and combine it with constraint equations to obtain the dynamic equations of the rotating MFC laminate. The specific method is as follows:

[0094] The mass matrix of the MFC laminate element is:

[0095] M AB =M A +(T BA ) T M B T BA (twenty three)

[0096] The stiffness matrix of the MFC laminate element is:

[0097] K AB =K A +(T BA ) T K MFC T BA +(T BA ) T K MFCE T BA , (twenty four)

[0098] In the formula, K A K MFC and K MFCE These are the stiffness matrix of the flexible plate element, the stiffness matrix of the MFC element, and the stiffness matrix of the electric field force, respectively.

[0099] The generalized external force matrix of the MFC laminate element is:

[0100] F aAB =F aA +(T BA ) T F aB T BA (25)

[0101] In the formula, F aA and F aB These are the generalized external force matrices acting on the flexible plate element and the MFC plate element, respectively.

[0102] By introducing the Boolean matrix B i Assume the absolute node coordinates of the i-th MFC laminate element are e i The absolute node coordinates of the MFC laminate elements and the node coordinates of the MFC laminate satisfy the following relationship:

[0103] e i =B i e (26)

[0104] In the formula, B i Let be a Boolean matrix, and e be the absolute node coordinates of the MFC laminate.

[0105] By introducing Boolean matrices, we can obtain the mass matrix M, stiffness matrix K, and generalized external force matrix F of the MFC laminate. a :

[0106]

[0107] In the formula M i K i F ai Let M be the mass matrix, stiffness matrix, and external force matrix of the i-th MFC laminate element, respectively. The presence of an MFC element is determined by its MFC attachment location; if so, then MFC is... i K i F ai Equal to M AB K AB F AB Otherwise M i K i F ai Equal to M A K A F A .

[0108] Therefore, the dynamic equation of the MFC laminate can be obtained as follows:

[0109]

[0110] In the formula ψ(ei ) is the nodal coordinate constraint equation under the constraint that the left end of the MFC laminate rotates around the Z-axis at a rotational speed w.

[0111] Step 5: Solve the dynamic equation of the rotating MFC laminate based on the generalized-α method to obtain the absolute node coordinates of the MFC laminate and calculate the position changes of the points on the MFC laminate. This will give us the parameters such as the deformation and velocity of the rotating MFC laminate.

[0112] Example

[0113] To verify the effectiveness of the present invention, the following embodiments were carried out.

[0114] The parameters for the rotating plate are set as shown in Table 1.

[0115] Table 1 Rotary Plate Parameter Settings

[0116]

[0117] Based on the absolute node coordinate method, a dynamic model of a rotating MFC laminate was performed, the dynamic equations were derived, and the algorithm program was written in MATLAB. The interface is shown below. Figure 3 As shown. After inputting the parameters, as... Figure 4 As shown, clicking the run button will generate a graph of the vibration response over time, which can accurately output the dynamic response at the end of the rotating plate over a period of time and allow for intuitive observation of the overall motion trajectory changes.

[0118] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0119] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A simulation method for vibration control response of a rotating MFC laminate, characterized in that, Includes the following steps: Step 1: Establish a physical model of the rotating MFC laminate, which consists of a flexible plate and an MFC plate, with the MFC plate located on the upper side of the flexible plate. Set the geometric parameters, material parameters, and motion parameters of the flexible plate, as well as the position parameters of the MFC plate. Step 2: Discretize the flexible plate and the MFC plate into elements, use the absolute node coordinate method to describe the motion and deformation of the flexible plate and the MFC plate, and combine the laminated plate element modeling method to obtain the coordinate transformation matrix between the flexible plate element and the upper MFC plate element. Step 3: Calculate the generalized elastic force and stiffness matrix of the flexible plate unit from the elastic potential energy of the flexible plate unit, introduce the constitutive equation of MFC to describe its electromechanical coupling characteristics, and use it as the actuator in the rotating MFC laminate to determine the generalized elastic force matrix, generalized stiffness matrix and generalized piezoelectric force matrix of the MFC plate unit, and obtain the control voltage parameters according to the PD control strategy. As the actuator in the MFC laminate, the MFC plate exhibits electromechanical coupling characteristics. Under the action of an external electric field, it will undergo structural deformation. Introducing the constitutive equation of the MFC plate into the flexible plate element model, its constitutive relation is expressed as: In the formula, σ B For the stress matrix of the MFC plate element, c B Let ε be the elastic coefficient matrix of the MFC plate element. B For the strain matrix of the MFC plate element, e E Let E be the piezoelectric coupling coefficient matrix of the MFC board element, D be the electric displacement of the MFC board element, ζ be the relative permittivity matrix of the MFC board element, and E be the piezoelectric coupling coefficient matrix of the MFC board element. e The magnitude of the electric field intensity applied to the MFC plate unit is: In the formula, U is the voltage applied to the MFC board unit, and h e The electrode spacing in an MFC board unit; Step 4: Obtain the element mass matrix, stiffness matrix and external force matrix of the MFC laminate element through coordinate transformation matrix, assemble the MFC laminate element, and obtain the dynamic equation of rotating MFC laminate by combining the constraint equation. Step 5: Solve the dynamic equation of the rotating MFC laminate based on the generalized-α method, calculate the absolute node coordinates of the MFC laminate, and obtain the deformation and velocity of the rotating MFC laminate.

2. The simulation method for vibration control response of a rotating MFC laminate according to claim 1, characterized in that, Step 1: Set the geometric parameters, material parameters, and motion parameters of the flexible board, as well as the position parameters of the MFC board. The specific method is as follows: (1) Parameters of flexible plates Geometric parameters: length L, width W, and thickness t; Material parameters: density ρ A Elastic modulus E and Poisson's ratio v; Motion parameters: Rotational speed w; (2) Parameters of MFC board Position parameters.

3. The simulation method for vibration control response of a rotating MFC laminate according to claim 2, characterized in that, Step 2: Discretize the flexible plate and MFC plate into elements. Use the absolute node coordinate method to describe the motion and deformation of the flexible plate and MFC plate. Combine the laminated plate element modeling method to obtain the coordinate transformation matrix between the flexible plate elements and the upper MFC plate elements. The specific method is as follows: The flexible plate and MFC plate are discretized into m elements along the length direction and n elements along the width direction, resulting in a finite number of flexible plate elements and MFC plate elements. A global coordinate system is established with the centroid of the left end face of the MFC laminate as the origin, the X-axis along the length direction as the x-axis, the inward width direction as the y-axis, and the upward thickness direction as the z-axis. A coordinate system for the flexible plate element is established with the midpoint of the outer vertical edge of the left end of the flexible plate element as the origin, the X-axis along the length direction as the x-axis, the inward width direction as the y-axis, and the upward thickness direction as the z-axis. The position r of any point P on the flexible plate element is represented in the global coordinate system as: Where, r x r y r z These are the coordinates of point P on the coordinate axes; The position of point P is represented by a shape function and absolute nodal coordinates as follows: r=[S1I S2I … S 16 I]e A =S A e A (2) Among them, e A The coordinates of the absolute element nodes of the flexible plate include 3 position coordinates and 9 partial derivatives of the position coordinates with respect to x, y, and z. I is a 3×3 identity matrix, S A The shape function of the flexible plate element is expressed as: In the formula, x, y and z are the coordinates of point P in the coordinate system of the flexible plate element, and a, b and t are the length, width and thickness of the flexible plate element, respectively. MFC laminated plate element modeling requires the following two basic assumptions to be met: 1) Before and after the deformation of the MFC laminate unit, the tangential planes of the upper and lower unit end faces are always coplanar; 2) During the motion and deformation process of the flexible plate unit and the MFC plate unit, the contact surface never slips relative to each other; Based on these two fundamental assumptions, the laminate consists of plate A and plate B, with plate B glued directly above plate A. M A M is the midpoint of the line connecting the two nodes on the left end of board A. B E is the midpoint of the line connecting the two nodes on the left end of board B. A For M A With M B The intersection of the line connecting the two sides and the upper left boundary of plate A, E B For M A With M B The intersection of the line connecting the two sides and the lower left boundary of plate B, M A M B E A E B The four points are always collinear, E A E B The two points always coincide, thus we get E. A E B The absolute nodal coordinates of the two points are equal: In the formula, E respectively A E B The positions of two points in the global coordinate system E respectively A E B The partial derivatives of the positions of two points in the global coordinate system with respect to x in their respective unit coordinate systems. E respectively A E B The partial derivatives of the positions of two points in the global coordinate system with respect to y in their respective unit coordinate systems. E respectively A E B The partial derivatives of the positions of two points in the global coordinate system with respect to z in their respective unit coordinate systems; E A With E B The absolute nodal coordinates of the two points are expressed using the shape functions and absolute nodal coordinates of the upper and lower plates, respectively: In the formula, S A S B These are the shape function matrices for the flexible board and the MFC board, respectively. Let x be the partial derivatives of the shape function matrices of the flexible board and the MFC board, respectively. Let be the partial derivatives of the shape function matrices of the flexible board and the MFC board with respect to y, respectively. Let z be the partial derivatives of the shape function matrices of the flexible plate and the MFC plate, respectively, with respect to z. E respectively A Point and E B absolute node coordinates of point e A and e B The partial derivative, e A and e B These are the absolute node coordinates of the flexible board element and the MFC board element, respectively. A t represents the thickness of the flexible plate unit. B The thickness of the MFC board unit; Substituting formulas (5) and (6) into (4), we get: Let F A F is the intersection of the midpoint of the line connecting the two nodes on the right end of plate A and the midpoint of the line connecting the two nodes on the right end of plate B, and the upper boundary on the right end of plate A. B G is the intersection of the midpoint of the line connecting the two nodes on the right end of plate A and the midpoint of the line connecting the two nodes on the right end of plate B, and the lower boundary of the right end of plate B; A G is the intersection of the midpoint of the line connecting the two nodes at the front end of board A and the midpoint of the line connecting the two nodes at the front end of board B, and the upper boundary of the front end of board A. B H is the intersection of the midpoint of the line connecting the two nodes at the front end of plate A and the midpoint of the line connecting the two nodes at the front end of plate B, and the lower boundary of the front end of plate B; A H is the intersection of the midpoint of the line connecting the two nodes at the rear end of plate A and the midpoint of the line connecting the two nodes at the rear end of plate B, and the upper boundary of the rear end of plate A. B The intersection of the midpoint of the line connecting the two nodes at the rear end of plate A and the midpoint of the line connecting the two nodes at the rear end of plate B with the lower boundary of the rear end of plate B. For point F A F B G A G B H A H B Similarly, there are: F respectively A Point and F B absolute node coordinates of point e A and e B The partial derivative, G A Point and G B absolute node coordinates of point e A and e B The partial derivative, H respectively A Point and H B absolute node coordinates of point e A and e B The partial derivative; Combining formulas (7) and (8), we get: In the formula, T BA This is the transformation matrix of absolute node coordinates between the upper and lower plate units.

4. The simulation method for vibration control response of a rotating MFC laminate according to claim 3, characterized in that, Step 3: Calculate the generalized elastic force and stiffness matrix of the flexible plate element from its elastic potential energy. Introduce the constitutive equation of the MFC (Mechanical-Fuel-Cellular) system to describe its electromechanical coupling characteristics. Use this equation as the actuator in the rotating MFC laminate to determine the generalized elastic force matrix, generalized stiffness matrix, and generalized piezoelectric force matrix of the MFC plate element. Obtain the control voltage parameters according to the PD (Power-Distributed Control) strategy. The specific method is as follows: Elastic potential energy E of flexible plate unit ε for: In the formula, σ A ε A V A c A These are the stress, strain, volume, and elastic coefficient matrices of a flexible plate element, respectively. A The expression is: In the formula, E is the elastic modulus of the flexible plate, and ν is the Poisson's ratio of the flexible plate. Taking the partial derivative of equation (10) with respect to the nodal coordinates of the flexible plate element, we obtain the generalized elastic force of the flexible plate element as follows: In the formula K A Here is the stiffness matrix of the flexible plate element; The generalized mass matrix of the flexible plate element is: The required control voltage is obtained based on the PD control strategy. The PD control method is as follows: In the formula, k p k is the proportional gain coefficient. d Let be the differential time constant, and error(t) be the displacement in the Z direction at the free end of the MFC laminate. The velocity in the Z direction at the free end of the MFC laminate; From the constitutive equation of the MFC plate element, the strain energy of the MFC plate element can be obtained as follows: In the formula V B This refers to the volume of an MFC board unit; The generalized elastic force of the MFC plate element is obtained by taking the partial derivative of the strain energy equation with respect to the absolute nodal coordinates of the MFC plate element. The stiffness matrix of the MFC plate element is obtained by taking the partial derivative of the generalized elastic force of the MFC plate element with respect to the nodal coordinates of the MFC plate element: The work done by the electric field force acting on the MFC board unit is: The generalized electric force is obtained by taking the partial derivative of the work done by the electric field force with respect to the nodal coordinates of the MFC plate element: In the formula K MFCE That is, the stiffness matrix of the electric field force of the MFC plate element; The generalized mass matrix of the MFC board element is 5. The simulation method for vibration control response of a rotating MFC laminate according to claim 4, characterized in that, Step 4: Obtain the element mass matrix, stiffness matrix, and external force matrix of the MFC laminate element through the coordinate transformation matrix. Assemble the MFC laminate element and, combined with the constraint equations, obtain the dynamic equations of the rotating MFC laminate. The specific method is as follows: The mass matrix of the MFC laminate element is: M AB =M A +(T BA ) T M B T BA (23) The stiffness matrix of the MFC laminate element is: K AB =K A +(T BA ) T K MFC T BA +(T BA ) T K MFCE T BA , (24) In the formula, K A K MFC and K MFCE These are the stiffness matrix of the flexible plate element, the stiffness matrix of the MFC element, and the stiffness matrix of the electric field force, respectively. The generalized external force matrix of the MFC laminate element is: F aAB =F aA +(T BA ) T F aB T BA , (25) In the formula, F aA and F aB These are the generalized external force matrices acting on the flexible plate element and the MFC plate element, respectively. By introducing the Boolean matrix B i Assume the absolute node coordinates of the i-th MFC laminate element are e i The absolute node coordinates of the MFC laminate element and the node coordinates of the MFC laminate satisfy the following relationship: e i =B i e (26) In the formula, B i Let be a Boolean matrix, and e be the absolute node coordinates of the MFC laminate. By introducing Boolean matrices, we obtain the mass matrix M, stiffness matrix K, and generalized external force matrix F of the MFC laminate. a : In the formula M i K i F ai Let M be the mass matrix, stiffness matrix, and external force matrix of the i-th MFC laminate element, respectively. The presence of an MFC element is determined by its MFC attachment location; if so, then MFC is... i K i F ai Equal to M AB K AB F AB Otherwise M i K i F ai Equal to M A K A F A ; Therefore, the dynamic equation of the MFC laminate is obtained as follows: In the formula ψ(e i ) is the nodal coordinate constraint equation under the constraint that the left end of the MFC laminate rotates around the Z-axis at a rotational speed w.

6. A simulation system for vibration control response of a rotating MFC laminate, used to implement the simulation method for vibration control response of a rotating MFC laminate as described in any one of claims 1-5, and to realize the simulation of vibration control response of the rotating MFC laminate.

7. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the simulation method for vibration control response of a rotating MFC laminate as described in any one of claims 1-5, thereby simulating the vibration control response of the rotating MFC laminate.

8. A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it implements the simulation method for vibration control response of a rotating MFC laminate as described in any one of claims 1-5, thereby realizing the simulation of vibration control response of a rotating MFC laminate.

Images