Dynamic response simulation method of a central rigid-flexible beam covered with piecewise constrained layer damping

Abstract

This paper discloses a method for simulating the dynamic response of a central rigid-flexible beam with a segmented constrained layer damping. The method simplifies a space manipulator into a central rigid-flexible beam structure, overlaying the segmented constrained layer damping on the central rigid-flexible beam. Based on floating coordinate theory, the finite element method is used for discretization. Considering high-order coupling deformation terms, a rigid-flexible coupled dynamic model of the central rigid-flexible beam with segmented constrained layer damping is established according to the second-kind Lagrangian equations. The dynamic equations of this system are solved using the generalized α-method, resulting in a lateral displacement-time curve at the end of the segmented constrained layer damping central rigid-flexible beam. This paper provides a new dynamic model for active and hybrid vibration control of flexible beam structures, providing researchers in this field with more comprehensive data and images.

Description

Technical Field

[0001] The invention relates to dynamic modeling of a flexible multi-body system, in particular to a dynamic response simulation method of a central rigid body-flexible beam system covered with segmented constrained layer damping. Background Art

[0002] Segmented constrained layer damping (SCD) is a vibration control technology that improves on traditional active constrained layer damping (ALD). It simultaneously cuts through both the piezoelectric constrained layer and the viscoelastic damping layer. This creates a concentrated region of shear deformation in the viscoelastic damping layer, thereby improving the structure's damping characteristics. Because the effectiveness of the segmented approach is significantly affected by parameters such as the placement of the cutouts and the thickness of the damping layer, it is crucial to dynamically model a central rigid-body-flexible beam system covered with SCD.

[0003] In his paper "Piecewise Constrained-Layer Damping Structure and Its Application in Vibration Reduction of Space Manipulators," Tian Shitao conducted dynamic modeling of a cantilever beam covered with a piecewise constrained-layer damper and studied the effectiveness and applicability of the piecewise approach. He noted that the piecewise approach is always applicable to the first-order mode of the structure, but for higher-order modes, it can be effective for extremely flexible structures. However, his study only analyzed the cantilever beam and did not consider the effects of rotation on the system's dynamic characteristics. Summary of the Invention

[0004] The present invention aims to provide a dynamic response simulation method for a central rigid body-flexible beam system covered with segmented constrained layer damping.

[0005] The technical solution for achieving the purpose of the present invention is: a method for simulating the dynamic response of a central rigid-flexible beam covered with piecewise constrained layer damping, comprising the following steps:

[0006] Step 1: Establish a physical model of a central rigid body-flexible beam covered with piecewise constrained layer damping, and set the material parameters, geometric parameters, and motion parameters of the model;

[0007] Step 2: Describe the deformation of the central rigid-flexible beam covered with the segmented constrained layer damping based on the floating coordinate system theory, and obtain the kinetic energy and potential energy of the central rigid-flexible beam covered with the segmented constrained layer damping;

[0008] Step 3: Using the finite element method to discretize the central rigid-flexible beam covered with piecewise constrained layer damping, the kinetic energy and potential energy of the discretized central rigid-flexible beam unit covered with piecewise constrained layer damping are obtained; at the same time, based on the proportional-differential control law, the generalized piezoelectric control force of the central rigid-flexible beam unit covered with piecewise constrained layer damping is obtained;

[0009] Step 4: Substitute the kinetic energy, potential energy, generalized piezoelectric control force, and driving torque of the central rigid-flexible beam unit covered with piecewise constrained layer damping into the second kind of Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the central rigid-flexible beam unit covered with piecewise constrained layer damping. By assembling the units, the overall dynamic equation of the central rigid-flexible beam covered with piecewise constrained layer damping is obtained.

[0010] Step 5: Use the generalized-α method to solve the dynamic equation of the entire central rigid body-flexible beam covered with piecewise constrained layer damping, and obtain the lateral displacement-time curve of the end of the central rigid body-flexible beam covered with piecewise constrained layer damping.

[0011] A simulation system for the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping realizes the simulation of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping based on the simulation method for the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping.

[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, the simulation of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping is realized based on the simulation method of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping.

[0013] A computer-readable storage medium stores a computer program. When the computer program is executed by a processor, the simulation method of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping is based on the simulation method of the dynamic response of the central rigid-flexible beam covered with segmented constrained layer damping.

[0014] Compared with the existing technology, the present invention has the following significant advantages: (1) The rotation of the central rigid-flexible beam covered with a segmented constrained layer damping beam is taken into account, and the dynamic characteristics of the central rigid-flexible beam covered with a segmented constrained layer damping beam are analyzed. (2) The high-order deformation coupling term is taken into account, and a new high-order rigid-flexible coupling dynamic model of the central rigid-flexible beam covered with a segmented constrained layer damping is established, which provides a certain reference and guidance for practical engineering applications. (3) Multi-physical field coupling is taken into account, including the piezoelectric effect of piezoelectric materials and the influence of driving torque. By changing the geometric parameters and material parameters of the structure, the dynamic response of the flexible beam under different conditions can be obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 Flowchart of the present invention.

[0016] Figure 2 Schematic diagram of a central rigid-flexible beam covered with piecewise constrained layer damping.

[0017] Figure 3 Schematic diagram of the deformation of the central rigid-flexible beam covered with piecewise constrained layer damping.

[0018] Figure 4 Comparison of the lateral deformation of the central rigid-flexible beam with piecewise constrained layer damping and active constrained layer damping in the open-loop case.

[0019] Figure 5 To control the influence of the lateral deformation at the end of a central rigid-flexible beam covered with piecewise constrained layer damping.

[0020] Figure 6 The lateral deformation of the central rigid-flexible beam covered with piecewise constrained layer damping varies with the proportional control gain k p 's change curve. DETAILED DESCRIPTION

[0021] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0022] like Figure 1 As shown, the present invention covers a dynamic response simulation method of a central rigid body-flexible beam system with segmented constrained layer damping, comprising the following steps:

[0023] Step 1: Establish a physical model of a central rigid-flexible beam covered with piecewise constrained layer damping, and set the material parameters, geometric parameters, and motion parameters of the model.

[0024] (1) Physical model of a central rigid-flexible beam with piecewise constrained layer damping

[0025] The central rigid body-flexible beam covered with segmented constrained layer damping includes a central rigid body and a flexible beam covered with segmented constrained layer damping. The flexible beam covered with segmented constrained layer damping consists of three sublayers: a base beam, a viscoelastic damping layer, and a piezoelectric constrained layer. The piezoelectric constrained layer composed of piezoelectric material is located at the top layer, the viscoelastic damping layer composed of viscoelastic material is located in the middle layer, and the base beam is the bottom layer. The segmented constrained layer damping is achieved by cutting the viscoelastic damping layer and the piezoelectric constrained layer at the same position.

[0026] (2) Material parameters, geometric parameters and motion parameters of the model

[0027] The material parameters are the elastic modulus E of each layer i , density ρ i , where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively, as well as the shear modulus G2 of the viscoelastic damping layer; the geometric parameters are the thickness of each layer h i, moment of inertia I i , where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively, as well as the beam length L, beam width b, radius R of the central rigid body, and the moment of inertia of the central rigid body J. oh The position of the cutout in the piecewise constrained layer damping is k, and the number of cutouts is n; the motion parameter is the driving torque F acting on the central rigid body τ .

[0028] Step 2: Based on the floating coordinate system theory, the deformation of the central rigid body-flexible beam covered with the segmented constrained layer damping is described to obtain the kinetic energy and potential energy of the central rigid body-flexible beam covered with the segmented constrained layer damping.

[0029] (1) Deformation of the central rigid-flexible beam covered with segmented constrained layer damping

[0030] With the center of the central rigid body as the origin o, the direction of the neutral axis of the base beam at point o is set as the x-axis, the direction perpendicular to the x-axis along the thickness of the base beam is set as the z-axis, and the direction perpendicular to the x-axis along the width of the base beam is set as the y-axis. The global coordinate system o-xyz of the central rigid body-flexible beam covered with segmented constrained layer damping is established. The axial deformation of the upper and lower ends of the viscoelastic damping layer near the central rigid body along the x-axis direction is expressed as u A and u B To express:

[0031]

[0032] where w is the lateral deformation of the central rigid-flexible beam covered with piecewise constrained layer damping, x is the abscissa of any point on the flexible beam, and u is i is the longitudinal deformation of each layer of the central rigid-flexible beam covered with piecewise constrained layer damping along the x-axis, where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively;

[0033] The axial deformation of the viscoelastic damping layer along the x-axis is:

[0034]

[0035] in,

[0036] The shear strain of the viscoelastic damping layer can be expressed as:

[0037]

[0038] in, The superscript “'” indicates the first-order partial derivative with respect to x.

[0039] (2) Kinetic and potential energy of the central rigid-flexible beam covered with segmented constrained layer damping

[0040] The position vector of any point on the central rigid-flexible beam covered with piecewise constrained layer damping can be expressed as:

[0041] r i =(R+x+u i )x+wz,i=1,2,3 (4)

[0042] The axial deformation u of each layer of the central rigid-flexible beam covered with piecewise constrained layer damping is i =w i +w c , w i represents the axial deformation of each layer of the central rigid-flexible beam covered with segmented constrained layer damping along the neutral axis, where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively, and w c represents the axial shortening deformation caused by the lateral bending of the central rigid-flexible beam covered with piecewise constrained layer damping, which can be written as the second-order coupled deformation term:

[0043]

[0044] in is the horizontal coordinate of any point on the central rigid-flexible beam covered with piecewise constrained layer damping.

[0045] The kinetic energy T of the central rigid-flexible beam covered with piecewise constrained layer damping and rotating along a fixed axis is s It can be expressed as:

[0046]

[0047] in, is the angular velocity of the central rigid body, A1, A2, and A3 are the cross-sectional areas of the base layer, viscoelastic damping layer, and piezoelectric constrained layer in the central rigid body-flexible beam covered with piecewise constrained layer damping, respectively. The superscript “·” indicates the first-order partial derivative with respect to time t.

[0048] The potential energy of the central rigid-flexible beam covered with segmented constrained layer damping is expressed as U=U1+U2+U3, where U1, U2 and U3 are the potential energies of the base beam, viscoelastic damping layer and piezoelectric constrained layer, respectively.

[0049] The potential energy of the base beam can be expressed as:

[0050]

[0051] The superscript “”” indicates the second-order partial derivative with respect to x.

[0052] The potential energy of the viscoelastic damping layer can be expressed as:

[0053]

[0054] Among them U shear is the shear strain energy of the viscoelastic damping layer:

[0055]

[0056] Among them G * is the shear modulus of the viscoelastic damping layer, and the complex constant modulus model is used to describe the material properties of the viscoelastic damping layer. * It can be expressed as:

[0057] G * =G2(1+η) (10)

[0058] where η is the loss factor of the viscoelastic damping layer.

[0059] The potential energy of the piezoelectric confinement layer can be expressed as:

[0060]

[0061] where e 31 represents the piezoelectric constant, E Z represents the electric field of the piezoelectric constrained layer along the z axis, ∈ 33 Represents the dielectric constant.

[0062] Step 3: Use the finite element method to discretize the central rigid body-flexible beam covered with segmented constrained layer damping, and obtain the kinetic energy and potential energy of the central rigid body-flexible beam unit covered with segmented constrained layer damping after discretization; at the same time, based on the proportional-differential control law, obtain the generalized piezoelectric control force of the central rigid body-flexible beam unit covered with segmented constrained layer damping.

[0063] (1) Kinetic and potential energy of the central rigid-flexible beam element covered with segmented constrained layer damping

[0064] The finite element method is used to discretize the central rigid-flexible beam covered with piecewise constrained layer damping, and the beam is divided into N units, each of which has a length of L. e Take the e-th unit and set the neutral axis of the base beam near the central rigid body of the e-th unit as the origin of the unit coordinate system. Will The direction of the point along the neutral axis of the base beam is set as Axis, perpendicular to Axis, along the direction of the base beam thickness is set to Axis, perpendicular to Axis, along the width of the base beam is set to The axis establishes the element coordinate system of the e-th element of the central rigid-flexible beam covered with piecewise constrained layer damping

[0065] The node displacement vector of the e-th element is expressed as: q e ={w 1j w 3j w j w j ′w 1k w 3k w k w k ′} T , where w 1j and w 1k They represent the axial deformation of the base beam along the neutral axis at the previous node and the next node of the e-th unit, respectively. 3j and w 3k denote the axial deformation of the piezoelectric constrained layer along the neutral axis at the previous node and the next node at the e-th unit, respectively. j and w k denote the lateral deformation of the central rigid-flexible beam covered with piecewise constrained layer damping at the previous and next nodes of the e-th element, respectively. The axial deformations w1, w2, w3 along the neutral axis, the lateral deformation w, the deformation rotation w′ about the x-axis, and the shear strain γ of the viscoelastic damping layer of each layer of the central rigid-flexible beam covered with piecewise constrained layer damping are expressed as interpolating polynomials of the nodal displacements using shape functions N1, N2, N3, N4, N5, and N6:

[0066] {w1 w2 w3 ww′γ} T ={N1 N2 N3 N4 N5 N6} T q (12)

[0067] Where q is the nodal displacement vector of the central rigid-flexible beam with piecewise constrained layer damping. The relationship between the nodal displacement vectors of the central rigid-flexible beam with piecewise constrained layer damping and the beam element can be expressed as:

[0068] q e =B e q (13)

[0069] Among them B e It is a Boolean matrix determined by the unit number e:

[0070] No.1 2 … e e+1 … N+1

[0071]

[0072] Above it is the position of the identity matrix I in the Boolean matrix;

[0073] In formula (12), N1, N2, N3, N4, N5 and N6 are the shape functions corresponding to w1, w2, w3, w, w′ and γ respectively. The relationship between N1, N2, N3, N4, N5 and N6 and their shape functions in the unit coordinate system can be expressed as follows: (e) =N1B e T , N2 (e) =N2B e T , N3 (e) =N3B e T , N4 (e) =N4B e T , N5 (e) =N5B e T , N6 (e) =N6B e T ,in:

[0074]

[0075] in,

[0076] The axial shortening deformation w caused by the lateral bending of the central rigid-flexible beam covered with segmented constrained layer damping c Finite element discretization is performed, and we have:

[0077]

[0078] in, It can be expressed as:

[0079]

[0080] Among them, N4 (j) =N4B j T .

[0081] Substituting Equations (12), (13), and (15) into Equation (4), we can obtain the axial deformation u of each layer of the e-th unit of the central rigid-flexible beam covered with segmented constrained layer damping along the x-axis direction: i And the lateral deformation w is:

[0082]

[0083] From formula (18), the deformation velocity of each layer of the e-th unit of the central rigid-flexible beam covered with piecewise constrained layer damping can be obtained as:

[0084]

[0085] Substituting Equations (18) and (19) into Equations (6), (7), (8), and (11), we can obtain the kinetic energy T of the central rigid-flexible beam element covered with piecewise constrained layer damping after discretization by the finite element method: s (e) and potential energy U (e) :

[0086]

[0087]

[0088] (2) Generalized piezoelectric control force of the central rigid-flexible beam element with segmented constrained layer damping

[0089] The work done by the piezoelectric effect of the piezoelectric confinement layer W p It can be expressed as:

[0090]

[0091] Where, ε c =u3′ is the elastic axial strain of the piezoelectric constrained layer, is the strain caused by the piezoelectric effect (d 31 represents the piezoelectric strain constant).

[0092] The control voltage φ acting on the piezoelectric confinement layer c It can be expressed as:

[0093]

[0094] Among them, k p is the proportional control gain, k d is the differential control gain, is the induced voltage of the sensor, A s is the surface area of the sensor, k 3t is the dielectric constant, k 31 represents the electromechanical coupling factor, g 31 represents the piezoelectric voltage constant.

[0095] From equations (22) and (23), the generalized piezoelectric control force of the central rigid-flexible beam element covered with piecewise constrained layer damping after discretization by the finite element method can be obtained as:

[0096]

[0097] Step 4: Substitute the kinetic energy, potential energy, generalized piezoelectric control force, and driving torque of the central rigid-flexible beam unit covered with piecewise constrained layer damping into the second kind of Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the central rigid-flexible beam unit covered with piecewise constrained layer damping. By assembling the units, the overall dynamic equation of the central rigid-flexible beam covered with piecewise constrained layer damping is obtained.

[0098] (1) Rigid-flexible coupled dynamic equations of the central rigid-flexible beam element covered with piecewise constrained layer damping

[0099] The kinetic energy T of the central rigid-flexible beam element covering the piecewise constrained layer damping s (e) , potential energy U (e) , generalized piezoelectric control force Q p (e) And the driving torque F τ Substituted into the second kind of Lagrange equation, we have:

[0100]

[0101] From this, the rigid-flexible coupling dynamic equation of the e-th unit of the central rigid-flexible beam covered with piecewise constrained layer damping can be obtained as follows:

[0102]

[0103] Among them, M 11 (e) 、M 12 (e) 、M 21 (e) and M 22 (e) is the mass matrix of the central rigid-flexible beam element covering the piecewise constrained layer damping, Q θ (e) and Q q (e) is the generalized force matrix of the central rigid-flexible beam element with piecewise constrained layer damping.

[0104] (2) The overall dynamic equations of the central rigid-flexible beam covered with piecewise constrained layer damping

[0105] First, the piezoelectric constraint layer and the viscoelastic layer covering the base beam are cut to perform a segmentation process. The cut is placed at the node k between the units. After the beam is segmented, the degree of freedom at the kth node is changed from the original {w1 (k) w3 (k) w (k) w′ (k)} expands to {w1 (k) w 3r(k) w 3l (k) w (k) w′ (k)}, where w 3r (k) represents the axial deformation of the piezoelectric constraint layer of the previous unit at the cut k along the neutral axis, w 3l (k) represents the axial deformation of the piezoelectric constrained layer of the next unit at the cutout k along the neutral axis. Correspondingly, the mass matrix M of the central rigid-flexible beam unit covering the segmented constrained layer damping is 11 (e) 、M 12 (e) 、M 21 (e) and M 22 (e) and the generalized force matrix Q of the central rigid-flexible beam element covered with piecewise constrained layer damping θ (e) and Q q (e) The size of q and M will also increase with the expansion of the degree of freedom at the cut position. After n cuts are applied to the beam, 12 (e) The size of M changes from 4(N+1)×1 to [4(N+1)+n]×1. 21 (e) and Q q (e) The size of M will increase from 1×4(N+1) to 1×[4(N+1)+n], 22 (e) The size of M will increase from 4(N+1)×4(N+1) to [4(N+1)+n]×[4(N+1)+n], 11 (e) 、 and Q θ (e) The size of remains unchanged at 1×1. After completing the unit assembly, the mass matrix and generalized force matrix of the central rigid body-flexible beam unit covered with piecewise constrained layer damping are added together to obtain the overall dynamic equation of the central rigid body-flexible beam covered with piecewise constrained layer damping:

[0106]

[0107] Among them, M 11 、M 12 、M 21 and M 22 is the mass matrix of the entire central rigid-flexible beam covering the piecewise constrained layer damping, Q θ and Q qis the generalized force matrix of the entire central rigid-flexible beam with piecewise constrained layer damping.

[0108] Step 5: Use the generalized-α method to solve the dynamic equation of the entire central rigid body-flexible beam covered with piecewise constrained layer damping, and obtain the lateral displacement-time curve of the end of the central rigid body-flexible beam covered with piecewise constrained layer damping.

[0109] Example

[0110] This embodiment calculates the dynamic response of a central rigid body-flexible beam system with piecewise constrained layer damping based on MATLAB. The specific method is as follows:

[0111] Step 1: In this embodiment, the central rigid body-flexible beam covered with segmented constrained layer damping adopts the geometric parameters and material parameters shown in Table 1. Set the driving torque F acting on the central rigid body τ for:

[0112]

[0113] Where τ0 is 1 N·m, the period T is 2 seconds, and t is taken as 0-3 seconds in the calculation, that is, 1.5 periods.

[0114] Table 1 Geometric and material parameters of the central rigid-flexible beam with segmented constrained layer damping

[0115]

[0116] Step 2: Based on the floating coordinate system theory, Figure 3 The deformation diagram of the central rigid-flexible beam covered with segmented constrained layer damping describes the deformation of the central rigid-flexible beam covered with segmented constrained layer damping, obtains the kinetic energy and potential energy of the central rigid-flexible beam covered with segmented constrained layer damping, and proceeds to step 3.

[0117] Step 3: Use the finite element method to discretize the central rigid body-flexible beam covered with segmented constrained layer damping, obtain the kinetic energy and potential energy of the central rigid body-flexible beam unit covered with segmented constrained layer damping after discretization, obtain the generalized piezoelectric control force of the central rigid body-flexible beam unit covered with segmented constrained layer damping based on the proportional-differential control law, and proceed to step 4.

[0118] Step 4: Based on the second-kind Lagrangian equation, establish the high-order rigid-flexible coupling dynamic equations of the central rigid body-flexible beam unit covered with piecewise constrained layer damping. Then, when assembling the unit, place the cutouts at the nodes between the two units to obtain the dynamic equations of the entire central rigid body-flexible beam covered with piecewise constrained layer damping, and proceed to step 5.

[0119] Step 5: Use the generalized-α method to solve the dynamic equations of the central rigid body-flexible beam covered with piecewise constrained layer damping. Use MATLAB programming to calculate the lateral displacement-time curve of the end of the central rigid body-flexible beam covered with piecewise constrained layer damping, and compare the lateral deformation of the end of the central rigid body-flexible beam covered with active constrained layer damping and piecewise constrained layer damping in the open loop case, as shown in Figure 5. Figure 4 As shown in the figure, it can be seen that the segmented method can reduce the maximum amplitude of the rotating beam, and when t>T, that is, when the driving torque of the structure is zero, the maximum amplitude of the SCLD rotating beam is also significantly smaller than the maximum vibration amplitude of the ACLD rotating beam. It can be seen that when the material parameters and size parameters shown in Table 1 are taken, the vibration suppression effect of the segmented active constrained layer damping is better than that of the simple active constrained layer damping. The proportional differential control law is used to analyze the lateral deformation of the central rigid-flexible beam covered with segmented constrained layer damping in the closed loop. Figure 5 k p =1,k d =-0.005, the influence of control on the lateral deformation of the beam end. It can be seen from the figure that after the control is applied to the SCLD beam, the amplitude of its vibration becomes smaller and the vibration suppression effect of the beam is also better. Therefore, applying control to the vibration suppression of the structure has a very obvious suppression effect. Figure 6 k d =0, k p The transverse deformation curve of the beam end under different conditions. It can be seen from the figure that with the proportional control coefficient k p The increase of k p When k = 5, the lateral deformation of the SCLD rotating beam is reduced to the maximum extent. p After it becomes larger on this basis, its lateral deformation is also reduced, but the extent of the reduction is smaller.

[0120] Based on previous research, this paper performs dynamic calculations on a central rigid body-flexible beam system covering the SCLD using MATLAB, providing a new dynamic model for active and hybrid vibration control of flexible beam structures and providing researchers in this field with more complete data and image information.

[0121] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, 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.

[0122] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present application shall be determined by the appended claims.

Claims

1. A dynamic response simulation method for a central rigid-flexible beam with segmented constrained layer damping, characterized in that: The following steps are involved: Step 1: Establish a physical model of a central rigid body-flexible beam covered with piecewise constrained layer damping, and set the material parameters, geometric parameters, and motion parameters of the model; Step 2: Describe the deformation of the central rigid-flexible beam covered with the segmented constrained layer damping based on the floating coordinate system theory, and obtain the kinetic energy and potential energy of the central rigid-flexible beam covered with the segmented constrained layer damping; Step 3: Using the finite element method to discretize the central rigid-flexible beam covered with piecewise constrained layer damping, the kinetic energy and potential energy of the discretized central rigid-flexible beam unit covered with piecewise constrained layer damping are obtained; at the same time, based on the proportional-differential control law, the generalized piezoelectric control force of the central rigid-flexible beam unit covered with piecewise constrained layer damping is obtained; Step 4: Substitute the kinetic energy, potential energy, generalized piezoelectric control force, and driving torque of the central rigid-flexible beam unit covered with piecewise constrained layer damping into the second kind of Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the central rigid-flexible beam unit covered with piecewise constrained layer damping. By assembling the units, the overall dynamic equation of the central rigid-flexible beam covered with piecewise constrained layer damping is obtained. Step 5: Use the generalized-α method to solve the dynamic equation of the entire central rigid body-flexible beam covered with piecewise constrained layer damping, and obtain the lateral displacement-time curve of the end of the central rigid body-flexible beam covered with piecewise constrained layer damping.

2. The method for simulating the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping according to claim 1, characterized in that: Step 1: Establish a physical model of a central rigid-flexible beam covered with piecewise constrained layer damping, and set the material parameters, geometric parameters, and motion parameters of the model. The specific method is as follows: (1) Physical model of a central rigid-flexible beam with piecewise constrained layer damping The central rigid body-flexible beam covered with segmented constrained layer damping includes a central rigid body and a flexible beam covered with segmented constrained layer damping. The flexible beam covered with segmented constrained layer damping consists of three sublayers: a base beam, a viscoelastic damping layer, and a piezoelectric constrained layer. The piezoelectric constrained layer composed of piezoelectric material is located at the top layer, the viscoelastic damping layer composed of viscoelastic material is located in the middle layer, and the base beam is the bottom layer. The segmented constrained layer damping is achieved by cutting the viscoelastic damping layer and the piezoelectric constrained layer at the same position. (2) Material parameters, geometric parameters and motion parameters of the model The material parameters are the elastic modulus E of each layer i , density ρ i , where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively, as well as the shear modulus G2 of the viscoelastic damping layer; the geometric parameters are the thickness of each layer h i , moment of inertia I i , where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively, as well as the beam length L, beam width b, radius R of the central rigid body, and the moment of inertia of the central rigid body J. oh The position of the cutout in the piecewise constrained layer damping is k, and the number of cutouts is n; the motion parameter is the driving torque F acting on the central rigid body τ .

3. The dynamic response simulation method of a central rigid-flexible beam covered with segmented constrained layer damping according to claim 2, characterized in that: Step 2: Based on the floating coordinate system theory, the deformation of the central rigid-flexible beam covered with the segmented constrained layer damping is described to obtain the kinetic energy and potential energy of the central rigid-flexible beam covered with the segmented constrained layer damping. The specific method is as follows: (1) Deformation of the central rigid-flexible beam covered with segmented constrained layer damping With the center of the central rigid body as the origin o, the direction of the neutral axis of the base beam at point o is set as the x-axis, the direction perpendicular to the x-axis along the thickness of the base beam is set as the z-axis, and the direction perpendicular to the x-axis along the width of the base beam is set as the y-axis. The global coordinate system o-xyz of the central rigid body-flexible beam covering the segmented constrained layer damping is established. Then the axial deformation u of the upper and lower ends of the viscoelastic damping layer near the central rigid body along the x-axis direction is A and u B for: where w is the lateral deformation of the central rigid-flexible beam covered with piecewise constrained layer damping, x is the abscissa of any point on the flexible beam, and u is i is the longitudinal deformation of each layer of the central rigid-flexible beam covered with piecewise constrained layer damping along the x-axis, where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively; The axial deformation of the viscoelastic damping layer along the x-axis is: in, The shear strain of the viscoelastic damping layer is: in, The superscript "'" indicates the first-order partial derivative of x; (2) Kinetic and potential energy of the central rigid-flexible beam covered with segmented constrained layer damping The position vector of any point on the central rigid-flexible beam covered with piecewise constrained layer damping is expressed as: r i =(R+x+u i )x+wz,i=1,2,3 (4) Among them, the axial deformation u of each layer of the central rigid-flexible beam covered with segmented constrained layer damping is i =w i +w c , w i represents the axial deformation of each layer of the central rigid-flexible beam covered with segmented constrained layer damping along the neutral axis, where the subscripts i = 1, 2, and 3 represent the base layer, viscoelastic damping layer, and piezoelectric constrained layer, respectively, and w c represents the axial shortening deformation caused by the lateral bending of the central rigid-flexible beam covered with piecewise constrained layer damping, and is written as the second-order coupled deformation term: where ζ is the abscissa of any point on the central rigid-flexible beam covered with piecewise constrained layer damping; The kinetic energy T of the central rigid-flexible beam covered with piecewise constrained layer damping and rotating along a fixed axis is s for: in, is the angular velocity of the central rigid body, A1, A2, and A3 are the cross-sectional areas of the base layer, viscoelastic damping layer, and piezoelectric constrained layer in the central rigid body-flexible beam covered with piecewise constrained layer damping, respectively. The superscript "·" indicates the first-order partial derivative with respect to time t; The potential energy of the central rigid-flexible beam covered with piecewise constrained layer damping is determined by the potential energies U1, U2, and U3 of the base beam, viscoelastic damping layer, and piezoelectric constrained layer, and is expressed as: U = U1 + U2 + U3, where: The potential energy of the base beam is: The superscript """ indicates the second-order partial derivative of x; The potential energy of the viscoelastic damping layer is: Among them U shear is the shear strain energy of the viscoelastic damping layer: Among them G * is the shear modulus of the viscoelastic damping layer, and the complex constant modulus model is used to describe the material properties of the viscoelastic damping layer. * Expressed as: G * =G2(1+η) (10) Where η is the loss factor of the viscoelastic damping layer; The potential energy of the piezoelectric confinement layer is: where e 31 represents the piezoelectric constant, E Z represents the electric field of the piezoelectric constrained layer along the z axis, ∈ 33 Represents the dielectric constant.

4. The method for simulating the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping according to claim 3, characterized in that: Step 3: The finite element method is used to discretize the central rigid-flexible beam covered with piecewise constrained layer damping, and the kinetic energy and potential energy of the discretized central rigid-flexible beam unit covered with piecewise constrained layer damping are obtained. At the same time, based on the proportional-differential control law, the generalized piezoelectric control force of the central rigid-flexible beam unit covered with piecewise constrained layer damping is obtained. The specific method is as follows: (1) Kinetic and potential energy of the central rigid-flexible beam element covered with segmented constrained layer damping The finite element method is used to discretize the central rigid-flexible beam covered with piecewise constrained layer damping, and the beam is divided into N units, each of which has a length of L. e , the node displacement vector of the e-th unit is expressed as: q e ={w 1j w 3j w j w j ′ w 1k w 3k w k w k ′} T , where w 1j and w 1k They represent the axial deformation of the base beam along the neutral axis at the previous node and the next node of the e-th unit, respectively. 3j and w 3k denote the axial deformation of the piezoelectric constrained layer along the neutral axis at the previous node and the next node at the e-th unit, respectively. j and w k denote the lateral deformation of the central rigid-flexible beam covered with piecewise constrained layer damping at the previous and next nodes of the e-th element, respectively. The shape functions N1, N2, N3, N4, N5, and N6 are used to express the axial deformations w1, w2, and w3 of each layer of the central rigid-flexible beam covered with piecewise constrained layer damping along the neutral axis, the lateral deformation w, the deformation rotation angle w′ around the x-axis, and the shear strain γ of the viscoelastic damping layer as interpolating polynomials of the nodal displacements: {w1 w2 w3 w w′ γ} T ={N1 N2 N3 N4 N5 N6} T q (12) Where q is the nodal displacement vector of the central rigid-flexible beam with piecewise constrained layer damping. The relationship between the nodal displacement vectors of the central rigid-flexible beam with piecewise constrained layer damping and the beam element is expressed as: q e =B e q (13) Among them, B e It is a Boolean matrix determined by the unit number e: Above it is the position of the identity matrix I in the Boolean matrix; In formula (12), N1, N2, N3, N4, N5 and N6 are the shape functions corresponding to w1, w2, w3, w, w′ and γ respectively. The relationship between N1, N2, N3, N4, N5 and N6 and their shape functions in the unit coordinate system can be expressed as follows: (e) =N1B e T , N2 (e) =N2B e T , N3 (e) =N3B e T , N4 (e) =N4B e T , N5 (e) =N5B e T , N6 (e) =N6B e T ,in: in, The axial shortening deformation w caused by the lateral bending of the central rigid-flexible beam covered with segmented constrained layer damping c Finite element discretization is performed, and we have: in, Expressed as: Among them, N4 (j) =N4B j T ; Substituting equations (12), (13), and (15) into equation (4), we can obtain the axial deformation u of each layer of the e-th unit of the central rigid-flexible beam covered with segmented constrained layer damping along the x-axis direction: i And the lateral deformation w is: The deformation velocity of each layer of the e-th unit of the central rigid-flexible beam covered with piecewise constrained layer damping is obtained from formula (18): Substituting Equations (18) and (19) into Equations (6), (7), (8), and (11), we can obtain the kinetic energy T of the central rigid-flexible beam element covered with piecewise constrained layer damping after discretization by the finite element method: s (e) and potential energy U (e) : (2) Generalized piezoelectric control force of the central rigid-flexible beam element with segmented constrained layer damping The work done by the piezoelectric effect of the piezoelectric confinement layer W p for: Where, ε c =u3′ is the elastic axial strain of the piezoelectric constrained layer, is the strain caused by the piezoelectric effect, d 31 represents the piezoelectric strain constant; The control voltage φ acting on the piezoelectric confinement layer c for: Among them, k p is the proportional control gain, k d is the differential control gain, is the induced voltage of the sensor, A s is the surface area of the sensor, k 3t is the dielectric constant, k 31 represents the electromechanical coupling factor, g 31 represents the piezoelectric voltage constant; From equations (22) and (23), the generalized piezoelectric control force of the central rigid-flexible beam element covered with piecewise constrained layer damping after discretization by the finite element method is obtained as follows:

5. The method for simulating the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping according to claim 4, characterized in that: Step 4: Substitute the kinetic energy, potential energy, generalized piezoelectric control force, and driving torque of the central rigid-flexible beam unit covered with piecewise constrained layer damping into the second-kind Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the central rigid-flexible beam unit covered with piecewise constrained layer damping. By assembling the units, the overall dynamic equation of the central rigid-flexible beam covered with piecewise constrained layer damping is obtained. The specific method is: (1) Rigid-flexible coupled dynamic equations of the central rigid-flexible beam element covered with piecewise constrained layer damping The kinetic energy T of the central rigid-flexible beam element covering the piecewise constrained layer damping s (e) , potential energy U (e) , generalized piezoelectric control force Q p (e) And the driving torque F τ Substituted into the second kind of Lagrange equation, we have: The rigid-flexible coupling dynamic equation of the e-th unit of the central rigid-flexible beam covered with piecewise constrained layer damping is obtained as follows: Among them, M 11 (e) 、M 12 (e) 、M 21 (e) and M 22 (e) is the mass matrix of the central rigid-flexible beam element covering the piecewise constrained layer damping, Q θ (e) and Q q (e) is the generalized force matrix of the central rigid-flexible beam element covered with piecewise constrained layer damping; (2) The overall dynamic equations of the central rigid-flexible beam covered with piecewise constrained layer damping The piezoelectric constraint layer and viscoelastic layer covering the base beam are cut to perform a segmentation process. The cut is placed at the node k between the units. After the beam is segmented, the degree of freedom at the kth node is changed from the original {w1 (k) w3 (k) w (k) w′ (k) } expands to {w1 (k) w 3r (k) w 3l (k) w (k) w′ (k) }, where w 3r (k) represents the axial deformation of the piezoelectric constraint layer of the previous unit at the cut k along the neutral axis, w 3l (k) represents the axial deformation of the piezoelectric constrained layer of the next unit at the cutout k along the neutral axis. Correspondingly, the mass matrix M of the central rigid-flexible beam unit covering the segmented constrained layer damping is 11 (e) 、M 12 (e) 、M 21 (e) and M 22 (e) and the generalized force matrix Q of the central rigid-flexible beam element covered with piecewise constrained layer damping θ (e) and Q q (e) The size of q and M will also increase with the expansion of the degree of freedom at the cut position. After n cuts are applied to the beam, 12 (e) The size of M changes from 4(N+1)×1 to [4(N+1)+n]×1. 21 (e) and Q q (e) The size of M will increase from 1×4(N+1) to 1×[4(N+1)+n], 22 (e) The size of M will increase from 4(N+1)×4(N+1) to [4(N+1)+n]×[4(N+1)+n], 11 (e) 、 and Q θ (e) The size of remains unchanged at 1×1; After completing the unit assembly, the mass matrix and generalized force matrix of the central rigid-flexible beam unit covered with piecewise constrained layer damping are added together to obtain the overall dynamic equation of the central rigid-flexible beam covered with piecewise constrained layer damping: Among them, M 11 、M 12 、M 21 and M 22 is the mass matrix of the entire central rigid-flexible beam covering the piecewise constrained layer damping, Q θ and Q q is the generalized force matrix of the entire central rigid-flexible beam with piecewise constrained layer damping.

6. A simulation system for the dynamic response of a central rigid-flexible beam with segmented constrained layer damping, characterized in that: Based on the simulation method of the dynamic response of the central rigid-flexible beam covered with segmented constrained layer damping according to any one of claims 1 to 5, the simulation of the dynamic response of the central rigid-flexible beam covered with segmented constrained layer damping is realized.

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, the simulation of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping is realized based on the simulation method of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping according to any one of claims 1 to 5.

8. A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the simulation of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping based on the simulation method of the dynamic response of a central rigid-flexible beam covered with segmented constrained layer damping as described in any one of claims 1 to 5.

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