Dynamic response simulation method of rotating flexible beam with partially covered ACLD based on SMC
Through the partially covered active constraining layer damping method based on sliding mode control, an accurate rotating flexible beam dynamic model is established, which solves the accuracy of vibration control in the prior art and achieves a better vibration suppression effect.
Patent Information
- Application Number
- CN202310367652.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-04-07
- Publication Date
- 2025-08-19
- Estimated Expiration
- 2043-04-07
AI Technical Summary
The prior art is difficult to accurately describe the deformation and vibration response of rotating flexible beams in vibration control, especially when combined with SMC control methods, the lack of accurate dynamic models leads to poor vibration suppression effects.
The partially covered active constraint layer damping method based on sliding mode control is adopted. The kinetic energy and potential energy of discretely rotating flexible beams are assumed by assuming the modal method, combined with the sliding mode control theory and the generalized-α method, an accurate rigid-flexible coupling dynamic equation is established, and vibration suppression is performed through the control force.
The precise deformation and vibration response simulation of rotating flexible beams is realized, which significantly improves the vibration suppression effect and provides a better control model for the rotating flexible beam structure.
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Figure CN116449729B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to a multi-body system dynamics modeling technology, in particular to a dynamic response simulation method of a rotating flexible beam with partially covered active constrained layer (ACLD) based on sliding mode control (SMC). Background Art
[0002] Vibration control of flexible multibody systems is a critical issue currently in need of improvement in engineering applications. Active constrained layer damping (ALD), a technique that combines passive constrained damping with pure active control, is gaining increasing attention. The design and construction of large spacecraft and high-speed rotating blades employ a variety of novel structures and materials to achieve lightweighting. This, in turn, introduces new vibrations, necessitating the development of novel technologies to suppress these detrimental vibrations.
[0003] In the paper "Dynamic Modeling and Analysis of a Rotating Flexible Beam with Smart ACLD Treatment," Li established a high-order rigid-flexible coupling dynamic model of a fully covered ACLD high-speed rotating flexible beam / plate by using a rigid-flexible coupling dynamic modeling method. This model simultaneously considers the rigid-flexible coupling of the system and the electromechanical coupling effects of the structure, but does not consider the SMC control method or other control methods used in this scheme to more effectively control the vibration of the structure. In "Research on Active Vibration and Attitude Control Methods for Rapid Maneuvering Processes of Flexible Spacecraft," Dong Chao proposed an improved sliding mode variable structure controller based on the sliding mode control principle, which can effectively reduce the residual vibration of the flexible attachment of the spacecraft. However, the design of this controller is not based on an accurate dynamic model and may not be accurate enough for related modeling and analysis. Summary of the Invention
[0004] The purpose of the present invention is to propose a dynamic response simulation method for a rotating flexible beam with partially covered ACLD based on SMC.
[0005] The technical solution to achieve the purpose of the present invention is: a dynamic response simulation method for a rotating flexible beam with a partially covered ACLD based on SMC, comprising the following steps:
[0006] Step 1: Set the geometric parameters, material parameters, and motion parameters of the rotating flexible beam system with partial coverage ACLD based on SMC;
[0007] Step 2, using a floating coordinate system method to determine the deformation of the rotating flexible beam based on the SMC partially covered ACLD, and obtain the kinetic energy and potential energy of the rotating flexible beam system based on the SMC partially covered ACLD;
[0008] Step 3: Discretize the rotating flexible beam based on the SMC partially covered ACLD using the assumed modal method to obtain the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD; substitute the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD into the second kind Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD;
[0009] Step 4: Based on the sliding mode control theory, the control force of the rotating flexible beam system with SMC-based partially covered ACLD is obtained. The control force is substituted into the rigid-flexible coupling dynamic equation of the rotating flexible beam system with SMC-based partially covered ACLD to obtain the dynamic equation of the rotating flexible beam system with SMC-based partially covered ACLD under closed-loop conditions.
[0010] Step 5: Use the generalized-α method to perform numerical simulation calculations on the system dynamics equations of the rotating flexible beam system based on the SMC partially covered ACLD, and obtain a lateral deformation-time curve diagram of the end of the rotating flexible beam based on the SMC partially covered ACLD.
[0011] A simulation system for the dynamic response of a rotating flexible beam with a partially covered ACLD based on SMC is provided. Based on the simulation method for the dynamic response of a rotating flexible beam with a partially covered ACLD based on SMC, the simulation of the dynamic response of the rotating flexible beam with a partially covered ACLD based on SMC is realized.
[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 rotating flexible beam with a partially covered ACLD based on SMC is realized based on the method for simulating the dynamic response of the rotating flexible beam with a partially covered ACLD based on SMC.
[0013] A computer-readable storage medium stores a computer program. When executed by a processor, the computer program simulates the dynamic response of a rotating flexible beam with a partially covered ACLD based on an SMC based on the method for simulating the dynamic response of the rotating flexible beam with a partially covered ACLD based on an SMC.
[0014] Compared with the prior art, the present invention has the following significant advantages: (1) Based on the central rigid body-cantilever beam model, a rigid-flexible coupling dynamic model of a rotating flexible beam with a partially covered ACLD based on SMC is established. This model can accurately describe the deformation of the rotating flexible beam with a partially covered ACLD based on SMC. In addition, a piezoelectric damping layer patch can be selectively covered on the flexible beam. By changing the parameter setting, the dynamic response of the displacement, deformation, and acceleration of the rotating flexible beam under different structures can be accurately calculated. (2) The SMC control method is introduced into the vibration control of the rotating flexible beam with a partially covered ACLD. The control force based on SMC is obtained through the precise rigid-flexible coupling dynamic equation. By using this control force to control the vibration of the rotating flexible beam with a partially covered ACLD based on SMC, a good suppression effect can be achieved, providing a new technical method for vibration suppression of ACLD structures. BRIEF DESCRIPTION OF THE DRAWINGS
[0015] Figure 1 It is a flow chart of the method of the present invention.
[0016] Figure 2 Schematic diagram of the rotating flexible beam structure with partially covered ACLD based on SMC.
[0017] Figure 3 The geometric deformation relationship diagram of the rotating flexible beam with partially covered ACLD based on SMC
[0018] Figure 4 It is the initialization diagram of the APP visual interface.
[0019] Figure 5 FIG. 1 is a diagram of the lateral deformation of the end of the rotating flexible beam of the partially covered ACLD based on SMC without control in the embodiment.
[0020] Figure 6 FIG. 1 is a diagram of the lateral deformation of the end of a rotating flexible beam based on a partially covered ACLD of SMC under SMC control in an embodiment. 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 dynamic response simulation method of the rotating flexible beam partially covered with ACLD based on SMC of the present invention includes the following steps:
[0023] Step 1: Set the geometric parameters, material parameters, and motion parameters of the rotating flexible beam system based on SMC partial coverage ACLD.
[0024] The rotating flexible beam system based on SMC partially covered ACLD consists of a rotating flexible beam and a central rigid body. Figure 2 As shown in the figure, the rotating flexible beam consists of a base beam layer and a covering patch, the covering patch is composed of two sub-layers, a viscoelastic damping layer and a piezoelectric constraint layer, and the covering patch and the base beam layer form a three-layer structure, which is called an ACLD structure.
[0025] (1) Geometric parameters
[0026] Geometric parameters of the base beam layer: the thickness of the base beam layer is h3, the width is b, and the length is l;
[0027] Geometric parameters of the patch: the thickness of the piezoelectric constraint layer is h1, the width is b, and the length is l p , the thickness of the viscoelastic damping layer is h2, the width is b, and the length is l p , the patch's coverage position is x j ;
[0028] Central rigid body geometric parameters: radius R, moment of inertia J h .
[0029] (2) Material parameters
[0030] Material parameters of the base beam layer: density is ρ3, elastic modulus is E3;
[0031] Material parameters of the patch: the density of the piezoelectric constrained layer is ρ1, the elastic modulus is E1, the density of the viscoelastic damping layer is ρ2, the elastic modulus is E2, and the shear modulus is G * =G2(1+η), where η is the loss factor of the viscoelastic damping layer.
[0032] (3) Motion parameters
[0033] The central rigid body is subjected to the external torque F τ Drive, and F τ for:
[0034] F τ =τexp(-120t) (1)
[0035] Here, τ is taken as 0.5 N·m, exp represents an exponential function with the natural constant e as the base, and t represents time.
[0036] Step 2: Use the floating coordinate system method to determine the deformation of the rotating flexible beam based on the SMC partially covered ACLD, and obtain the kinetic energy and potential energy of the rotating flexible beam system based on the SMC partially covered ACLD.
[0037] (1) Determining the deformation of a rotating flexible beam with a partially covered ACLD based on SMC
[0038] The coordinate system of the rotating flexible beam system with partial ACLD based on SMC is established as follows: Figure 3 As shown, in the principal coordinate system, the center of the central rigid body is taken as the origin O of the principal coordinate system, and the direction of the neutral axis of the base beam at the initial position at point O is set as the X-axis, which is perpendicular to the X-axis, and the direction along the thickness of the base beam is set as the Z-axis; in the floating coordinate system, the intersection of the central rigid body and the base beam is taken as the origin o of the floating coordinate system, and the direction of the neutral axis of the base beam at point O is set as the x-axis, which is perpendicular to the x-axis, and the direction along the thickness of the base beam is set as the z-axis.
[0039] Considering the influence of nonlinear coupling deformation, the axial deformation of the piezoelectric constrained layer and the base beam layer along the x-axis can be expressed as:
[0040] u1=w1(x,t)+w c (x, t) (2)
[0041] u3=w3(x,t)+w c (x, t) (3)
[0042] Among them, u i represents the axial deformation of the rotating flexible beam at point x along the x-axis at time t, w i (x, t) represents the deformation of each layer of the rotating flexible beam along the x-axis at time t, i = 1, 2, 3 correspond to the piezoelectric constraint layer, viscoelastic damping layer and base beam layer respectively, w c (x, t) represents the axial shortening of the rotating flexible beam caused by the lateral deformation of the rotating flexible beam at time t, that is, the nonlinear coupling deformation term, which is expressed as:
[0043]
[0044] Wherein, w in Equation (4) is the lateral deformation of the rotating flexible beam of the partially covered ACLD based on SMC along the z-axis, and ξ is the horizontal coordinate of any point on the rotating flexible beam of the partially covered ACLD based on SMC;
[0045] The axial deformation u of the upper and lower ends of the viscoelastic damping layer near the central rigid body along the x-axis a 、u b It can be expressed as:
[0046]
[0047]
[0048] The axial deformation of the viscoelastic damping layer along the x-axis can be expressed as:
[0049] u2=(u a +u b) / 2=(u1+u3+d1w′) / 2 (7)
[0050] Where d1 = (h1 - h3) / 2, and the superscript "'" indicates the first-order partial derivative with respect to the axial coordinate x;
[0051] The shear strain of the viscoelastic damping layer can be expressed as:
[0052]
[0053] Wherein, d0=(h1+2h2+h3) / 2.
[0054] (2) Kinetic energy and potential energy of the rotating flexible beam system with partially covered ACLD based on SMC
[0055] The position vector of any point on the rotating flexible beam of the SMC-based partial coverage ACLD can be expressed as:
[0056] r i =(R+x+u i )x+wz,i=1,2,3 (9)
[0057] The kinetic energy E of the rotating flexible beam with partial ACLD based on SMC rotating around the fixed axis is k It can be expressed as:
[0058]
[0059] in, is the angular velocity of the central rigid body, the superscript “·” indicates the first-order partial derivative with respect to time t, x j is the coverage position of the patch, A i is the cross-sectional area of each layer of the beam;
[0060] Its potential energy U can be expressed as:
[0061]
[0062] Among them, I i is the moment of inertia of each layer of the rotating flexible beam. The superscript “”” indicates the second partial derivative with respect to the axial coordinate x.
[0063] Step 3. Use the assumed modal method to discretize the rotating flexible beam based on the SMC partially covered ACLD to obtain the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD. Substitute the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD into the second kind of Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD. The specific method is as follows:
[0064] (1) Discrete kinetic and potential energies of a rotating flexible beam system with partially covered ACLD based on SMC
[0065] The rotating flexible beam is discretized using the assumed modal method. As the generalized coordinates of the rotating flexible beam system of the SMC-based partially covered ACLD, where θ represents the rotation angle of the central rigid body, q 1u ,q 3u ,q w denote the modal coordinate vectors of the longitudinal vibration and the transverse vibration in the rotating flexible beam system with partially covered ACLD based on SMC, respectively, and T denotes the transposed sign;
[0066] According to the assumed modal method, the axial and lateral deformations of the i-th layer after discretization in the rotating flexible beam with partially covered ACLD based on SMC can be expressed as:
[0067]
[0068] Among them, φ iu (x)∈R 1×N (i=1,3) and φ w (x)∈R 1×N are the row vectors of the mode functions of the axial and lateral vibrations of the rotating flexible beam with partially covered ACLD based on SMC, respectively; q iu (t)∈R N (i=1, 3) and q w (t)∈R N are the modal coordinate column vectors of the axial and lateral vibrations of the rotating flexible beam with SMC-based partially covered ACLD, respectively;
[0069] Substituting Equation (12) into Equations (2), (3), and (7), we can obtain the axial deformation of each layer of the discretized rotating flexible beam of the partially covered ACLD based on SMC:
[0070]
[0071] in, is the second-order coupled shape function;
[0072] From this, the deformation velocity of each layer of the discretized rotating flexible beam with partial coverage ACLD based on SMC can be obtained:
[0073]
[0074] Substituting Equations (13) and (14) into Equations (10) and (11), we can obtain the potential energy and kinetic energy of the rotating flexible beam system with partially covered ACLD based on SMC after discretization by the assumed modal method:
[0075]
[0076]
[0077] (2) The rigid-flexible coupling dynamic equation of the rotating flexible beam system with partial ACLD covered by SMC is discretized into the kinetic energy of the rotating flexible beam system with partial ACLD covered by SMC. potential energy Driving torque F τ , the control force u is substituted into the second kind of Lagrange equation, and we have:
[0078]
[0079] The rigid-flexible coupling dynamic equation of the rotating flexible beam system with partially covered ACLD based on SMC is obtained as follows:
[0080]
[0081] The generalized mass matrix of the rotating flexible beam with partially covered ACLD based on SMC is M, and the generalized force matrix is Q, and their expressions are:
[0082]
[0083] Step 4. Based on the sliding mode control theory, the control force of the rotating flexible beam system based on the SMC partially covered ACLD is obtained, and the control force is substituted into the rigid-flexible coupling dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD to obtain the dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD under closed-loop conditions.
[0084] (1) Vibration control force of a rotating flexible beam system with partially covered ACLD based on SMC
[0085] The rigid-flexible coupling dynamic equation of the rotating flexible beam system with partial ACLD based on SMC in Equation (18) ignores the effect of the rotation angle θ and is expanded into the following form:
[0086]
[0087] Among them,
[0088] The rigid-flexible coupling dynamic equation shown in Equation (19) is rewritten as a state-space equation:
[0089]
[0090] in, Y is a 2n×1 dimensional matrix, A is a 2n×2n dimensional matrix, B is a 2n×m dimensional matrix, and D represents the position matrix of the piezoelectric force as an n×m dimensional matrix, and u represents the control force of the piezoelectric layer under the control voltage as a 2n×1 dimensional matrix, which is expressed as:
[0091]
[0092] Among them, d 31 represents the piezoelectric strain constant, φ c Indicates the control voltage;
[0093] Assume that the sampling time of the sample is T d , k represents the kth sampling time, and the control force u within the sample time
[0094] Remain unchanged, specifically expressed as follows:
[0095] u(t)=u(kT d ),kT d ≤t≤(k+1)T d (twenty two)
[0096] Applying the zero-order hold method to the state-space equation (20) yields the following equation:
[0097]
[0098] Among them, t is the current time, t0 is the previous time;
[0099] Let t0 = kT d and t = (k + 1)T d , then the discretized form of the state space equation (20) can be obtained:
[0100] Y((k+1)T d )=A d (T d )Y(kT d )+B d (T d )u(kT d ) (twenty four)
[0101] in,
[0102] Simplify formula (24) to:
[0103] Y(k+1)=A d Y(k)+B d u(k) (25)
[0104] in, A d11 ∈R (2n-m)×(2n-m), A d12 ∈R (2n-m)×m , A d21 ∈R m×(2n-m) , A d22 ∈R m ×m , B d1 ∈R (2n-m)×m , B d2 ∈R m×m All are reversible;
[0105] By normalizing Equation (25) and introducing the transposed matrix T, we can obtain the normalized equation as follows:
[0106] Z(k)=TY(k) (26)
[0107] in,
[0108] Substituting Equation (26) into Equation (25) yields the canonical discrete state equation:
[0109]
[0110] in,
[0111] Construct the linear sliding surface of the rotating flexible beam system with partially covered ACLD based on SMC:
[0112] S(k)=GY(k) (28)
[0113] Where G = [G1 G2], G1∈R m×n , G2∈R m×n ;
[0114] Substituting formula (26) into formula (28) yields:
[0115]
[0116] in, Assuming G2 is the identity matrix, then
[0117]
[0118] The optimal cost function of the rotating flexible beam system with partial coverage ACLD based on SMC is as follows:
[0119]
[0120] Where Q is the weight matrix of the optimal value function of the rotating flexible beam system with partially covered ACLD based on SMC;
[0121] Substituting Equation (27) into (31), we can obtain the feedback matrix of the state space equation of the rotating flexible beam system based on SMC partial coverage ACLD:
[0122]
[0123] The P in the formula is obtained by the Riccati equation as follows:
[0124]
[0125] In the above formula:
[0126] Then you can get
[0127]
[0128] Substituting Equation (34) into Equation (28), the sliding mode surface of the rotating flexible beam system with partially covered ACLD based on SMC can be determined;
[0129] When the rotating flexible beam system based on SMC and partially covered ACLD enters the sliding surface, the following are obtained:
[0130] S(k+1)-S(k)=0 (35)
[0131] Substituting Equation (28) into Equation (35), the control force of the rotating flexible beam system with partially covered ACLD based on SMC can be obtained:
[0132] u eq =-(GB d ) -1 (G(A d -I)Y(k)) (36)
[0133] (2) Dynamic equations of the rotating flexible beam system based on SMC partially covered ACLD under closed-loop conditions
[0134] The control force u obtained by formula (36) eq Substituting this into the rigid-flexible coupling dynamic equation (18) of the rotating flexible beam system with partial coverage ACLD based on SMC, the dynamic equation of the rotating flexible beam system with partial coverage ACLD based on SMC under closed-loop conditions is obtained:
[0135]
[0136] Step 5: Use the generalized-α method to perform numerical simulation calculations on the system dynamics equations of the rotating flexible beam system with partial coverage ACLD based on SMC, and obtain the lateral deformation-time curve of the end of the rotating flexible beam with partial coverage ACLD based on SMC.
[0137] Example
[0138] This embodiment is based on the dynamic response simulation method of the rotating flexible beam partially covered by the ACLD of the SMC. The specific method is as follows:
[0139] Step 1. In this embodiment, the base beam layer, viscoelastic damping layer, and piezoelectric constraining layer of the SMC-based partially covered ACLD rotating flexible beam are configured using the parameters shown in Table 1. In this embodiment, the patch length is 0.1 m, the attachment position is 0.1 m from the central rigid body, and the thickness of the viscoelastic material layer is 1 mm.
[0140] Table 1 Material and geometric parameters of the partially covered ACLD rotating flexible beam system based on SMC
[0141]
[0142]
[0143] Step 2: According to Figure 3 The geometric deformation relationship diagram of the rotating flexible beam with partial coverage ACLD based on SMC is shown. The floating coordinate system method is used to describe the deformation of the rotating flexible beam with partial coverage ACLD based on SMC. The kinetic energy and potential energy of the rotating flexible beam system with partial coverage ACLD based on SMC are obtained, and then the process goes to step 3.
[0144] Step 3. Use the assumed modal method to discretize the rotating flexible beam based on the SMC partially covered ACLD to obtain the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD; substitute the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD into the second-kind Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD, and then proceed to step 4.
[0145] Step 4. Based on the sliding mode control theory, the control force of the rotating flexible beam system based on the SMC partially covered ACLD is obtained. The control force is substituted into the rigid-flexible coupling dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD. The dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD under closed-loop conditions is obtained, and then proceed to step 5.
[0146] Step 5: Use the generalized-α method to perform numerical simulation calculations on the system dynamics equations of the rotating flexible beam system based on the SMC partially covered ACLD, and obtain the lateral deformation-time curve of the end of the rotating flexible beam based on the SMC partially covered ACLD. By comparing the lateral deformation-time curve of the end of the rotating flexible beam based on the SMC partially covered ACLD without any control (such as Figure 5) and the lateral deformation-time curve of the end of the rotating flexible beam with partial ACLD covered by SMC control (as shown in Figure 6 As shown in Figure 3, it can be found that the lateral vibration suppression effect of the rotating flexible beam under SMC control is obvious and the suppression efficiency is very high.
[0147] Based on previous research, this embodiment introduces SMC control into the vibration control of the ACLD structure. The relevant dynamics of the system are calculated using MATLAB, and a schematic diagram of the lateral deformation of the flexible beam end is obtained. This provides a new control model for the vibration control of the rotating flexible beam structure, which has a better control effect on the vibration suppression of the rotating flexible beam structure.
[0148] 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.
[0149] 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 rotating flexible beam with partially covered ACLD based on SMC, characterized by: The steps include: Step 1: Set the geometric parameters, material parameters, and motion parameters of the rotating flexible beam system with partial coverage ACLD based on SMC; Step 2, using a floating coordinate system method to determine the deformation of the rotating flexible beam based on the SMC partially covered ACLD, and obtain the kinetic energy and potential energy of the rotating flexible beam system based on the SMC partially covered ACLD; Step 3: Discretize the rotating flexible beam based on the SMC partially covered ACLD using the assumed modal method to obtain the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD; substitute the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD into the second kind Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD; Step 4: Based on the sliding mode control theory, the control force of the rotating flexible beam system with SMC-based partially covered ACLD is obtained. The control force is substituted into the rigid-flexible coupling dynamic equation of the rotating flexible beam system with SMC-based partially covered ACLD to obtain the dynamic equation of the rotating flexible beam system with SMC-based partially covered ACLD under closed-loop conditions. Step 5, using the generalized-α method to perform numerical simulation calculations on the system dynamics equations of the rotating flexible beam system based on the SMC partially covered ACLD, and obtain a lateral deformation-time curve diagram of the end of the rotating flexible beam based on the SMC partially covered ACLD; Step 1: Set the geometric parameters, material parameters, and motion parameters of the rotating flexible beam system with partial ACLD based on SMC. The specific method is as follows: The SMC-based partially covered ACLD rotating flexible beam system consists of a rotating flexible beam and a central rigid body. The rotating flexible beam consists of a base beam layer and a covering patch. The covering patch is composed of two sub-layers, a viscoelastic damping layer and a piezoelectric constraint layer. The covering patch and the base beam layer form a three-layer structure, called the ACLD structure. (1) Geometric parameters Geometric parameters of the base beam layer: the thickness of the base beam layer is h3, the width is b, and the length is l; Geometric parameters of the patch: the thickness of the piezoelectric constraint layer is h1, the width is b, and the length is l p , the thickness of the viscoelastic damping layer is h2, the width is b, and the length is l p , the patch's coverage position is x j ; Central rigid body geometric parameters: radius R, moment of inertia J h ; (2) Material parameters Material parameters of the base beam layer: density is ρ3, elastic modulus is E3; Material parameters of the patch: the density of the piezoelectric constrained layer is ρ1, the elastic modulus is E1, the density of the viscoelastic damping layer is ρ2, the elastic modulus is E2, and the shear modulus is G * =G2(1+η), where η is the loss factor of the viscoelastic damping layer; (3) Motion parameters The central rigid body is subjected to the external torque F τ Drive, and F τ for: F τ =τexp(-120t) (1) Here, τ is taken as 0.5 N·m, exp represents an exponential function with the natural constant e as the base, and t represents time.
2. The dynamic response simulation method of a rotating flexible beam with partially covered ACLD based on SMC according to claim 1 is characterized in that: Step 2: Determine the deformation of the rotating flexible beam with partial ACLD based on SMC using the floating coordinate system method, and obtain the kinetic energy and potential energy of the rotating flexible beam system with partial ACLD based on SMC. The specific method is as follows: (1) Determining the deformation of a rotating flexible beam with a partially covered ACLD based on SMC The coordinate system of the rotating flexible beam system with partially covered ACLD based on SMC is established. In the principal coordinate system, the center of the central rigid body is taken as the origin O of the principal coordinate system. The direction of the neutral axis of the base beam at the initial position at point O is set as the X-axis, which is perpendicular to the X-axis. The direction along the thickness of the base beam is set as the Z-axis. In the floating coordinate system, the intersection of the central rigid body and the base beam is taken as the origin o of the floating coordinate system. The direction of the neutral axis of the base beam at point O is set as the x-axis, which is perpendicular to the x-axis. The direction along the thickness of the base beam is set as the z-axis. Considering the influence of nonlinear coupling deformation, the axial deformation of the piezoelectric constraint layer and the base beam layer along the x-axis is expressed as follows: u1=w1(x,t)+w c (x,t) (2) u3=w3(x,t)+w c (x,t) (3) Among them, u i represents the axial deformation of the rotating flexible beam at point x along the x-axis at time t, w i (x, t) represents the deformation of each layer of the rotating flexible beam along the x-axis at time t, i = 1, 2, 3 correspond to the piezoelectric constraint layer, viscoelastic damping layer and base beam layer respectively, w c (x, t) represents the axial shortening of the rotating flexible beam caused by the lateral deformation of the rotating flexible beam at time t, that is, the nonlinear coupling deformation term, which is expressed as: Wherein, w in Equation (4) is the lateral deformation of the rotating flexible beam of the partially covered ACLD based on SMC along the z-axis, and ξ is the horizontal coordinate of any point on the rotating flexible beam of the partially covered ACLD based on SMC; The axial deformation u of the upper and lower ends of the viscoelastic damping layer near the central rigid body along the x-axis a 、u b It can be expressed as: The axial deformation of the viscoelastic damping layer along the x-axis can be expressed as: u2=(u a +u b ) / 2=(u1+u3+d1w′) / 2 (7) Where d1 = (h1 - h3) / 2, and the superscript "'" indicates the first-order partial derivative with respect to the axial coordinate x. The shear strain of the viscoelastic damping layer is expressed as: Where, d0 = (h1 + 2h2 + h3) / 2; (2) Kinetic energy and potential energy of the rotating flexible beam system with partially covered ACLD based on SMC The position vector of any point on the rotating flexible beam of the SMC-based partial coverage ACLD is expressed as: r i =(R+x+u i )x+wz,i=1,2,3 (9) The kinetic energy E of the rotating flexible beam with SMC-based partially covered ACLD rotating around the fixed axis is k Expressed as: in, is the angular velocity of the central rigid body, the superscript "·" indicates the first-order partial derivative with respect to time t, x j is the coverage position of the patch, A i is the cross-sectional area of each layer of the beam; Its potential energy U is expressed as: Among them, I i is the moment of inertia of each layer of the rotating flexible beam. The superscript """ indicates the second partial derivative with respect to the axial coordinate x.
3. The dynamic response simulation method of a rotating flexible beam with partially covered ACLD based on SMC according to claim 2, characterized in that: Step 3: Use the assumed modal method to discretize the rotating flexible beam based on the SMC partially covered ACLD to obtain the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD. Substitute the kinetic energy and potential energy of the discrete rotating flexible beam system based on the SMC partially covered ACLD into the second-kind Lagrangian equation to obtain the rigid-flexible coupling dynamic equation of the rotating flexible beam system based on the SMC partially covered ACLD. The specific method is as follows: (1) Discrete kinetic and potential energies of a rotating flexible beam system with partially covered ACLD based on SMC The rotating flexible beam is discretized using the assumed modal method. As the generalized coordinates of the rotating flexible beam system of the SMC-based partially covered ACLD, where θ represents the rotation angle of the central rigid body, q 1u ,q 3u ,q w denote the modal coordinate vectors of the longitudinal vibration and the transverse vibration in the rotating flexible beam system with partially covered ACLD based on SMC, respectively, and T denotes the transposed sign; According to the assumed modal method, the axial and lateral deformations of the i-th layer after discretization in the rotating flexible beam with partially covered ACLD based on SMC are expressed as: Among them, φ iu (x)∈R 1×N (i=1,3) and φ w (x)∈R 1×N are the row vectors of the mode functions of the axial and lateral vibrations of the rotating flexible beam with partially covered ACLD based on SMC, respectively; q iu (t)∈R N (i=1,3) and q w (t)∈R N are the modal coordinate column vectors of the axial and lateral vibrations of the rotating flexible beam with SMC-based partially covered ACLD, respectively; Substituting Equation (12) into Equations (2), (3), and (7), we can obtain the axial deformation of each layer of the discretized rotating flexible beam of the partially covered ACLD based on SMC: in, is the second-order coupled shape function; The deformation velocity of each layer of the discretized rotating flexible beam with partial coverage ACLD based on SMC is obtained as follows: Substituting Equations (13) and (14) into Equations (10) and (11), we can obtain the potential energy and kinetic energy of the rotating flexible beam system with partially covered ACLD based on SMC after discretization by the assumed modal method: (2) Rigid-flexible coupling dynamic equations of a rotating flexible beam system with partially covered ACLD based on SMC The kinetic energy of the rotating flexible beam system based on the SMC partial coverage ACLD after discretization potential energy Driving torque F τ , the control force u is substituted into the second kind of Lagrange equation, and we have: The rigid-flexible coupling dynamic equation of the rotating flexible beam system with partially covered ACLD based on SMC is obtained as follows: The generalized mass matrix of the rotating flexible beam with partially covered ACLD based on SMC is M, and the generalized force matrix is Q, and their expressions are:
4. The dynamic response simulation method of a rotating flexible beam with partially covered ACLD based on SMC according to claim 3 is characterized in that: Step 4: Based on the sliding mode control theory, the control force of the rotating flexible beam system with partial coverage ACLD based on SMC is obtained. The control force is substituted into the rigid-flexible coupling dynamic equation of the rotating flexible beam system with partial coverage ACLD based on SMC to obtain the dynamic equation of the rotating flexible beam system with partial coverage ACLD based on SMC under closed-loop conditions. The specific method is as follows: (1) Vibration control force of a rotating flexible beam system with partially covered ACLD based on SMC The rigid-flexible coupling dynamic equation of the rotating flexible beam system with partial ACLD based on SMC in Equation (18) ignores the effect of the rotation angle θ and is expanded into the following form: Among them, The rigid-flexible coupling dynamic equation shown in Equation (19) is rewritten as a state-space equation: in, Y is a 2n×1 dimensional matrix, A is a 2n×2n dimensional matrix, B is a 2n×m dimensional matrix, and D represents the position matrix of the piezoelectric force as an n×m dimensional matrix, and u represents the control force of the piezoelectric layer under the control voltage as a 2n×1 dimensional matrix, which is expressed as: Among them, d 31 represents the piezoelectric strain constant, φ c Indicates the control voltage; Assume that the sampling time of the sample is T d , k represents the kth sampling time, and the control force u remains unchanged within the sample time, which is specifically expressed as follows: u(t)=u(kT d ),kT d ≤t≤(k+1)T d (22) Applying the zero-order hold method to the state-space equation (20) yields the following equation: Among them, t is the current time, t0 is the previous time; Let t0 = kT d and t = (k + 1)T d , then the discretized form of the state space equation (20) can be obtained: Y((k+1)T d )=A d (T d )Y(kT d )+B d (T d )u(kT d ) (24) in, Simplify formula (24) to: Y(k+1)=A d Y(k)+B d u(k) (25) Among them, A d11 ∈R (2n-m)×(2n-m) , A d12 ∈R (2n-m)×m , A d21 ∈R m×(2n-m) , A d22 ∈R m×m , B d1 ∈R (2n-m)×m , B d2 ∈R m×m Uniform reversible; By normalizing Equation (25) and introducing the transposed matrix T, we can obtain the normalized equation as follows: Z(k)=TY(k) (26) in, And B d1 ∈R (2n-m)×m , B d2 ∈R m×m All are reversible; Substituting Equation (26) into Equation (25) yields the canonical discrete state equation: in, Construct the linear sliding surface of the rotating flexible beam system with partially covered ACLD based on SMC: S(k)=GY(k) (28) Where G = [G1 G2], G1∈R m×n , G2∈R m×n ; Substituting formula (26) into formula (28) yields: in, Assuming G2 is the identity matrix, then The optimal cost function of the rotating flexible beam system with partial coverage ACLD based on SMC is as follows: Where Q is the weight matrix of the optimal value function of the rotating flexible beam system with partially covered ACLD based on SMC; Substituting Equation (27) into (31), we obtain the feedback matrix of the state space equation of the rotating flexible beam system with partial coverage ACLD based on SMC: The P in the formula is obtained by the Riccati equation as follows: In the above formula: Then we get Substituting Equation (34) into Equation (28), the sliding mode surface of the rotating flexible beam system with partially covered ACLD based on SMC can be determined; When the rotating flexible beam system based on SMC and partially covered ACLD enters the sliding surface, the following are obtained: S(k+1)-S(k)=0 (35) Substituting Equation (28) into Equation (35), we can obtain the control force of the rotating flexible beam system with partially covered ACLD based on SMC: you eq =-(GB d ) -1 (G(A d -I)Y(k)) (36) (2) Dynamic equations of the rotating flexible beam system based on SMC partially covered ACLD under closed-loop conditions The control force u obtained by formula (36) eq Substituting this into the rigid-flexible coupling dynamic equation (18) of the rotating flexible beam system with partial coverage ACLD based on SMC, the dynamic equation of the rotating flexible beam system with partial coverage ACLD based on SMC under closed-loop conditions is obtained:
5. A dynamic response simulation system for a rotating flexible beam with partially covered ACLD based on SMC, characterized in that: Based on the dynamic response simulation method of the rotating flexible beam with partially covered ACLD based on SMC according to any one of claims 1 to 4, the dynamic response simulation of the rotating flexible beam with partially covered ACLD based on SMC is realized.
6. 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 method for simulating the dynamic response of a rotating flexible beam with a partially covered ACLD based on an SMC is used to simulate the dynamic response of the rotating flexible beam with a partially covered ACLD based on an SMC, based on any one of claims 1 to 4.
7. A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, the method for simulating the dynamic response of a rotating flexible beam with a partially covered ACLD based on an SMC is based on any one of claims 1 to 4, thereby simulating the dynamic response of a rotating flexible beam with a partially covered ACLD based on an SMC.