A neural network-based intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers

Abstract

This invention discloses an intelligent active control method for the nonlinear dynamics of viscoelastic dielectric elastomers based on neural networks. The invention proposes a viscoelastic dynamics model for a dielectric elastomer, establishes a neural network model based on the numerical solutions obtained from simulations, and then establishes active control systems for free vibration and forced vibration respectively on the data-driven neural network model, achieving the effects of vibration suppression and tracking the desired output. The intelligent active control method proposed in this invention can be used in the design of active control systems using intelligent soft materials such as dielectric elastomers as actuators or sensors.

Description

Technical Field

[0001] This invention relates to the field of data-driven model establishment and active control of nonlinear dynamics of dielectric elastomers, specifically to an intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers based on neural networks. Background Technology

[0002] Dielectric elastomers are flexible smart materials that naturally deform under external stimuli, exhibiting expansion in area and contraction in thickness. They consist of two elastic materials sandwiched between flexible electrodes. The basic working principle is that when a voltage is applied through the two electrodes, the elastomer contracts along the electric field direction and expands laterally. Under aerodynamic pressure and high pressure, the maximum surface strain can exceed 2200%, a mechanism demonstrated by Rengen over a century ago. This enormous electrically driven strain immediately attracted the development of various applications for dielectric elastomers, recognizing them as one of the most promising artificial muscle technologies. Compared to other driving technologies such as piezoelectric, electrostrictive, magnetostrictive, and shape memory materials, dielectric elastomers offer advantages such as low density, low modulus, large driving strain, high driving speed, and high specific energy density. Dielectric elastomers have wide applications in biomimetic artificial muscles, soft robots, tunable lenses, tactile interfaces, and energy harvesting.

[0003] Given the rapid development of dielectric elastomers in various engineering fields, further quantitative research is needed on their dynamic characteristics and precise control of dynamic output. Accurate control of their dynamic response is crucial. Two aspects require attention: the dynamic governing equations of viscohyperelastic dielectric elastomers and active control methods for the nonlinear vibrations of dielectric elastomers. Dielectric elastomers possess properties close to natural muscle, making them a promising type of soft actuator. Establishing a dynamic model of viscohyperelasticity will allow for a better understanding of the dynamic response of dielectric elastomers and enable precise control in applications. In particular, nonlinear dynamic control is a significant aid in further mastering the development of this material in various fields.

[0004] To date, no data-driven model of dielectric elastomers has been studied using neural networks and applied to controller design. To fill this gap, this invention establishes a data-driven neural network model to fit the nonlinear dynamic vibration behavior of viscoelastic dielectric elastomers, and combines this model with a closed-loop control system to achieve vibration suppression and signal tracking. The method proposed in this invention provides a new approach for rapid modeling and active control design of smart soft materials such as dielectric elastomers, and provides a theoretical foundation for the further development of data-driven neural network methods in the constitutive direction of elastic soft materials. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention provides an intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers based on neural networks. It fully considers the difficulties in model establishment and active control commonly encountered in practical engineering problems, providing a feasible design method for active control systems using dielectric elastomers as actuators or sensors, resulting in designs with greater engineering applicability. Based on the physical model and the vibration characteristics of dielectric elastomers, this invention introduces a nonlinear autoregressive neural network with external input, providing a feasible design method for active control systems using dielectric elastomers as actuators or sensors, building upon a closed-loop control system.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A neural network-based intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers is proposed for the design of active control systems using dielectric elastomers as actuators or sensors. The method includes the following steps:

[0008] Step 1: Derive the viscoelastic constitutive model based on Maxwell's rheological model;

[0009] Step 2: Based on the principle of virtual work, obtain the governing equations for the planar rectangular dielectric elastomer film;

[0010] Step 3: Derive the nonlinear dynamic differential equations based on the Gent hyperelastic constitutive model;

[0011] Step 4: Based on the data obtained from the simulation of the nonlinear mechanics model, a neural network model is constructed using machine learning algorithms.

[0012] Step 5: Based on the nonlinear active control method, the neural network model is used to suppress nonlinear vibration and track the desired output effect of the dielectric elastomer vibration response.

[0013] Furthermore, the first step includes:

[0014] The viscoelasticity of the dielectric elastic body is represented by the Maxwell rheological model, consisting of two nonlinear springs and a damper. The two nonlinear springs are modeled by a hyperelastic model, with the two material parameters being the shear modulus μ and the tensile limit J. m The damper is characterized by viscosity η, and the deformation of the elastic body is described by the stretching in the X and Y directions: λ1 and λ2 are the stretching of the α spring in the rheological model; the stretching of the β spring is λ. β 1 and λ β 2. The damper's tension in the X and Y directions is ξ1 and ξ2, respectively;

[0015] The nominal stress tensor S is obtained by introducing the strain energy density function. 11 ,S 22 :

[0016]

[0017] Where σ1 is the normal stress in the X direction and σ2 is the normal stress in the Y direction.

[0018] Using a rheological model, the free energy function considering the viscoelasticity of a dielectric elastomer is obtained as follows:

[0019] W(λ1,λ2,ξ1,ξ2,D)=W α stretch +W β stretch +W electric (2)

[0020] Where D is the actual electric displacement in the elastic body, and W α stretch W represents the elastic potential energy generated by the deformation of an elastic dielectric in the X direction. β stretch W represents the elastic potential energy generated by the deformation of an elastic dielectric in the Y direction. electric It represents the electric potential energy of an elastic dielectric when it is deformed.

[0021] Furthermore, the second step includes:

[0022] Taking a dielectric elastomer film as the research object, the initial dimensions of the dielectric elastomer film are length 2L1, width 2L2, and thickness 2H in the X, Y, and Z directions, respectively. The dielectric elastomer film is pre-stretched by external forces P1 and P2 in the X and Y directions, respectively. When a voltage is applied to the thickness of the elastomer through two flexible electrodes on the top and bottom surfaces of the elastomer, the dielectric elastomer film expands its area in the XY plane and decreases its thickness in the Z direction. The dimensions of the deformed film are expressed as length 2l1, width 2l2, and thickness 2h.

[0023] The elongation of the film in the X, Y, and Z directions is uniform, expressed as λ1 = l1 / L1, λ2 = l2 / L2, and λ3 = h / H, respectively. Assuming the elastic body is incompressible, we have λ1λ2λ3 = 1. To apply the principle of virtual work, virtual displacement δu(X,Y) and virtual electric displacement δD are introduced. Both are spatially smooth and time-independent. X and Y are the components of the virtual displacement vector in the X and Y directions, respectively. The virtual work done by the external force P is:

[0024] δW1 ext =P1δu1(L1,Y)+P2δu2(X,L2)-P1δu1(-L1,Y)-P2δu2(X,L2) (3)

[0025] Where P1 and P2 are defined external forces in the in-plane direction, and body forces are ignored here;

[0026] The virtual work done by the external voltage is:

[0027]

[0028] Where Φ is the applied voltage, and δQ is the virtual increment of charge on the two electrodes;

[0029] Let D represent the actual electric displacement in the elastic body, which is related to the total change, as shown below:

[0030]

[0031] For uniform biaxial loading, the internal virtual work generated by material deformation is expressed as follows:

[0032] δW1 int =∫δF:SdV0=8δF ij :S ij L1L2L3=8L1L2L3(δλ1S 11 +δλ2S 22 (6)

[0033] Where S is the nominal stress tensor, F is the deformation gradient, and V0 is the initial state volume, i.e., λ1=λ2=λ3=1;

[0034] The internal virtual work generated by the electric field in the material is written as:

[0035]

[0036] Where ε is its own dielectric constant;

[0037] Virtual inertial work or virtual kinetic energy is:

[0038]

[0039] Where ρ0 is the density of the elastic dielectric elastomer;

[0040] In equilibrium, we have:

[0041] δW int -δW ext +δW kin =0 (9)

[0042] For uniform biaxial deformation, we have:

[0043]

[0044] Treating δλ1, δλ2, and δD as independent variables, and simplifying the equations, we obtain the governing equations for the dielectric elastomer film under uniform biaxial deformation:

[0045]

[0046]

[0047] Furthermore, the third step includes;

[0048] The strain energy function of the Gent hyperelastic model is as follows:

[0049]

[0050] Where J is the volume ratio, I1 is the corresponding first strain tensor invariant; μ is the initial shear modulus, and J m D1 is the maximum average deformation parameter, and D1 is the material's inherent incompressibility parameter.

[0051] Combining the incompressibility formula λ1λ2λ3=1, and the ideal dielectric elastic body model:

[0052]

[0053] get:

[0054]

[0055] Therefore, the dynamic governing equations are derived based on the constitutive model:

[0056]

[0057]

[0058]

[0059]

[0060] in, Let λ1 be the second derivative with respect to time. λ² is the second derivative with respect to time;

[0061] Set up using the prescribed dynamic model and Where η is viscosity;

[0062]

[0063]

[0064] Considering the case of biaxial deformation, i.e., λ1=λ2=λ, ξ1=ξ2=ξ, P1=P2=P and L1=L2=L, at the instant of initial vibration, the damper does not have enough time to deform, therefore ξ1(t=0)=ξ2(t=0)=1; as time progresses, the damper changes with time; the time-varying deformation of the elastic body λ(T) is calculated from the equilibrium equations with initial conditions:

[0065]

[0066]

[0067] After obtaining the above system of equations, solve the equations.

[0068] Furthermore, the fourth step includes;

[0069] Based on the above physical model and the numerical solution of the response obtained from simulation, a data-driven free vibration model of the dielectric elastic body is constructed. The neural network used is a NARX nonlinear discrete system model, expressed as:

[0070] y(t)=f{u(tD u ),…,u(t-1),u(t),y(tD y ),…,y(t-1)} (23)

[0071] In equation (23): u(t) and y(t) represent the input and output values ​​of the network at time t, respectively; where D u The maximum order of the input time extension steps; D y Let be the maximum order of the output time extension step; therefore, u(t-Du),...,u(t-1) are all stored inputs relative to time t; y(t-Dy),...,y(t-1) are stored inputs relative to time t; f is a nonlinear function obtained by fitting through a neural network; where the next value of the subordinate output signal y(t) is regressed based on the previous value of the output signal and the previous value of the independent input signal; where As input and output values, the hidden layer is set to 3 layers, with 10 neurons in each layer, and is trained. Then, data is prepared for validation to obtain a data-driven neural network model; the neural network is retrained multiple times each time it is trained.

[0072] Furthermore, the fifth step includes;

[0073] After obtaining the NARX model, a PID control algorithm is used to control the nonlinear dynamic neural network model of the dielectric elastomer. In the closed-loop system, for the PID controller:

[0074] The deviation between the given value and the actual output is expressed as:

[0075] e(t)=r(t)-c(t) (24)

[0076] Where r(t) is a given value, and c(t) is the actual output value of the system;

[0077] The control law is

[0078] The transfer function is now:

[0079] In the formula K P K is the proportionality coefficient. I K is the integration time constant. D Let K be the differential time constant, and U(s) and E(s) be the Laplace transforms of the output and input, respectively. P ,K I ,K D Debugging is performed according to the control objectives;

[0080] For the active control structure of the dielectric elastomer parameter vibration system, the active excitation is the applied voltage. The applied voltage is adjusted by the PID controller and fed into the neural network model. The neural network model generates a real-time feedback signal and achieves closed-loop control through PID control. To meet the different needs of suppressing vibration and tracking the desired output, the parameters are adjusted to achieve the control effect.

[0081] Beneficial effects:

[0082] The advantages of this invention compared to existing technologies lie in that it provides a new approach to establishing and controlling dynamic models of unknown elastic materials, overcoming and improving the limitations of traditional methods for establishing dynamic physical models, and making full use of experimental data. In practice, considering the need to simplify the process and avoid unnecessary resource waste, a data-based dynamic model is obtained by training a neural network model, allowing for a prior assessment of the control effect and reducing the drawbacks to the overall research caused by excessively high peak values ​​in control. Attached Figure Description

[0083] Figure 1 It is the viscoelastic Maxwell rheological model of dielectric elastomers;

[0084] Figure 2(a) is a schematic diagram of the initial state of the planar dielectric elastomer;

[0085] Figure 2(b) is a schematic diagram of the current state of the planar dielectric elastomer;

[0086] Figure 3 It is the NARX neural network model;

[0087] Figure 4 It is a PID control neural network closed-loop system;

[0088] Figure 5 These are numerical simulation results of the nonlinear dynamic model of a dielectric elastomer at different voltage frequencies;

[0089] Figure 6 These are the actual values ​​and neural network fitted values ​​under forced vibration;

[0090] Figure 7 It is a time-series diagram showing the changes in the vibration expansion and contraction rate before and after control.

[0091] Figure 8 This is a comparison chart of expected output and actual output;

[0092] Figure 9 This is a flowchart of the neural network-based intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers according to the present invention. Detailed Implementation

[0093] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the protection scope of the present invention.

[0094] This invention proposes an intelligent active control method for the nonlinear dynamics of viscoelastic dielectric elastomers based on neural networks, such as... Figure 9 As shown, it includes the following steps:

[0095] (1) Deriving the viscoelastic constitutive model based on Maxwell's rheological model. This invention uses Maxwell's rheological model to represent the viscoelasticity of dielectric elastomers, such as... Figure 1 As shown, it consists of two nonlinear springs and a damper. The two nonlinear springs are modeled using a hyperelastic model, with the two material parameters being the shear modulus μ and the tensile limit J. m The damper is characterized by a viscosity η, and the deformation of the elastomer is described by the stretching in the X and Y directions: λ1 and λ2, which are also the stretching of the α spring in the rheological model. The stretching of the β spring is λβ1 and λβ2, and the stretching of the damper in the X and Y directions is ξ1 and ξ2.

[0096] By introducing the strain energy density function, the nominal stress tensor S can be obtained. 11 ,S 22 :

[0097]

[0098] Where σ1 is the normal stress in the X direction and σ2 is the normal stress in the Y direction;

[0099] Using a rheological model, the free energy function considering the viscoelasticity of a dielectric elastomer is obtained as follows:

[0100] W(λ1,λ2,ξ1,ξ2,D)=W α stretch +W β stretch +W electric (2)

[0101] Where D represents the actual electric displacement in the elastic body, the specific meaning of which will be explained below, and W... α stretch W represents the elastic potential energy generated by the deformation of an elastic dielectric in the X direction. β stretch W represents the elastic potential energy generated by the deformation of an elastic dielectric in the Y direction. electric The potential energy of an elastic dielectric during deformation:

[0102] (2) The governing equations for a planar rectangular dielectric elastomer film are derived based on the principle of virtual work. The research object of this invention is a dielectric elastomer film. As shown in Figure 2(a), the initial dimensions of the dielectric elastomer film are length 2L1, width 2L2, and thickness 2H in the X, Y, and Z directions, respectively. The dielectric elastomer film is pre-stretched by external forces P1 and P2 in the X and Y directions, respectively. When a voltage is applied to the thickness of the elastomer through two flexible electrodes (compatible electrodes) on the top and bottom surfaces of the elastomer, the dielectric elastomer film expands its area in the XY plane and decreases its thickness in the Z direction. As shown in Figure 2(b), the dimensions of the deformable film are expressed as length 2l1, width 2l2, and thickness 2h.

[0103] The elongation of the film in the X, Y, and Z directions is uniform, expressed as λ1 = l1 / L1, λ2 = l2 / L2, and λ3 = h / H, respectively. Assuming the elastic body is incompressible, we have λ1λ2λ3 = 1. To apply the principle of virtual work, we introduce virtual displacement δu(X, Y) (where X and Y are the components of the virtual displacement vector in the X and Y directions, respectively) and virtual electric displacement δD, which are spatially smooth and time-independent. The virtual work done by the external force P is:

[0104] δW1 ext =P1δu1(L1,Y)+P2δu2(X,L2)-P1δu1(-L1,Y)-P2δu2(X,L2) (3)

[0105] Where P1 and P2 are defined external forces in the in-plane direction, and body forces are neglected here. The virtual work done by the external voltage is:

[0106]

[0107] Where Φ is the applied voltage, and δQ is the virtual increment of charge on the two electrodes. Let D represent the actual electric displacement in the elastic body, which is related to the total change, as shown below:

[0108]

[0109] For uniform biaxial loading, the internal virtual work generated by material deformation is expressed as follows:

[0110] δW1 int =∫δF:SdV0=8δF ij :S ij L1L2L3=8L1L2L3(δλ1S 11 +δλ2S 22 (6)

[0111] Where S is the nominal stress tensor, F is the deformation gradient, and V0 is the initial state volume (i.e., λ1 = λ2 = λ3 = 1);

[0112] The internal virtual work generated by the electric field in the material can be written as:

[0113]

[0114] Virtual inertial work or virtual kinetic energy is:

[0115]

[0116] In equilibrium, we have:

[0117] δW int -δW ext +δW kin =0 (9)

[0118] For uniform biaxial deformation, we have:

[0119]

[0120] Treating δλ1, δλ2, and δD as independent variables, and simplifying the equations, we obtain the governing equations for the dielectric elastomer film under uniform biaxial deformation:

[0121]

[0122]

[0123] (3) The strain energy function of the Gent hyperelastic model is as follows:

[0124]

[0125] Where J is the volume ratio, I1 is the corresponding first strain tensor invariant, and μ is the initial shear modulus.m D1 is the maximum average deformation parameter, and D1 is the material's inherent incompressibility parameter.

[0126] Combining the incompressibility formula λ1λ2λ3=1, and the ideal dielectric elastic body model:

[0127]

[0128] We can obtain:

[0129]

[0130] Therefore, the dynamic governing equations are derived based on the constitutive model:

[0131]

[0132]

[0133]

[0134]

[0135] in, Let λ1 be the second derivative with respect to time. λ² is the second derivative with respect to time;

[0136] Set up using the prescribed dynamic model and Where η is viscosity;

[0137]

[0138]

[0139] Considering the case of biaxial deformation, i.e., λ1=λ2=λ, ξ1=ξ2=ξ, P1=P2=P and L1=L2=L, at the instant of initial vibration, the damper does not have enough time to deform, therefore ξ1(t=0)=ξ2(t=0)=1; as time progresses, the damper changes with time; the time-varying deformation of the elastic body λ(T) is calculated from the equilibrium equation with initial conditions:

[0140]

[0141]

[0142] After obtaining the above system of equations, solve the equations.

[0143] (4) Based on the data obtained from the simulation of the nonlinear mechanics model, a machine learning algorithm is used to learn and construct a neural network model. Next, based on the above physical model and the numerical solution of the simulated response, a data-driven free vibration model of the dielectric elastic body is constructed, using the NARX nonlinear discrete system model. The NARX neural network incorporates delay and feedback mechanisms, enhancing its ability to remember historical data; it is a dynamic neural network. NARX is suitable for time series forecasting and has been applied to solve nonlinear sequence forecasting problems in various fields. It is represented as:

[0144] y(t)=f{u(tD u ),…,u(t-1),u(t),y(tD y ),…,y(t-1)} (23)

[0145] In equation (23): u(t) and y(t) represent the input and output values ​​of the network at time t, respectively; where D u The maximum order of the input time extension steps; D y The maximum order of the output time extension step is given; therefore, u(t-Du),...,u(t-1) are stored inputs relative to time t; y(t-Dy),...,y(t-1) are stored inputs relative to time t; f is a nonlinear function obtained by fitting through a neural network. For example... Figure 3 As shown, the next value of the subordinate output signal y(t) is regressed based on the previous value of the output signal and the previous value of the independent (external) input signal, where... The hidden layers are set to 3, with 10 neurons per layer. The model is trained using the `trainlm` function in MATLAB, and then validated using the `preparets` function to obtain a data-driven neural network model. Each time the neural network is trained, different initial weights and biases, as well as different ways of dividing the data into training, validation, and test sets, may produce different solutions. Therefore, different neural networks trained for the same problem may produce different outputs for the same input. To ensure the selection of a neural network with good accuracy, multiple retraining sessions are necessary.

[0146] (5) Based on the nonlinear active control method, the neural network model is used to suppress nonlinear vibration and track the desired output effect of the dielectric elastomer vibration response. For example... Figure 4 As shown, after obtaining the NARX model, this invention uses a PID control algorithm (Proportional Integral-Derivative Control algorithm) to control the nonlinear dynamic neural network model of the dielectric elastic body. In the closed-loop system, for the PID controller:

[0147] The deviation between the given value and the actual output is expressed as:

[0148] e(t)=r(t)-c(t) (24)

[0149] Where r(t) is a given value, and c(t) is the actual output value of the system;

[0150] The control law is

[0151] The transfer function is now:

[0152] In the formula K P K is the proportionality coefficient. I K is the integration time constant. D Let K be the differential time constant, and U(s) and E(s) be the Laplace transforms of the output and input, respectively. P ,K I ,K D It can be adjusted according to the control objective. For the active control structure of the dielectric elastomer parametric vibration system, the active excitation is the applied voltage. The applied voltage is adjusted by the PID controller and fed into the neural network model. The neural network model generates a real-time feedback signal, which is then used to achieve closed-loop control through PID control. To meet different needs such as vibration suppression and tracking of the desired output, the parameters are adjusted to achieve the control effect.

[0153] Example:

[0154] To gain a fuller understanding of the characteristics of this invention and its applicability to engineering practice, this invention verifies the proposed interval-based uncertain quasi-static analysis method for creep and relaxation conditions in quasi-static problems. Subsequently, to verify the proposed interval-based uncertain nonlinear dynamic analysis method, this invention performs a numerical simulation example for an in-plane deformable dielectric elastic body, achieving vibration suppression and tracking of the desired output, respectively.

[0155] In the embodiment of the quasi-static problem, the material parameters are μ1 = 18000 Pa, μ2 = 42000 Pa, J lim1 =110, J lim2 =55, ρ0=960kg / m 3 ε=3.9825×10 -11F / m; dimensional parameters are L1=L2=L=1m, H=3mm. With load P=100N and DC voltage=10kV, the equilibrium stretch ratio λ=3.8. Starting from the relaxed state, applying the initial conditions λ(0)=3.7, dλ(0) / dt=0 and ξ(0)=λ(0)=3.7, the dynamic response of DE was calculated using the RKDP method in the Runge-Kutta series ODE solver, and implemented in MATLAB using the ODE45 solver, where the sinusoidal excitation is: Φ(t)=Φ DC +Φ AC cos(2πf·t), where DC is the DC voltage, AC is the amplitude of the AC voltage, and f is the frequency of the AC voltage. We set DC = AC = 5kV. The initial condition is set to be in a fully relaxed state of DE under DC voltage, i.e., ξ(0) = λ(0) = 3.7, and the nonlinear vibration response of the viscoelastic dielectric elastic body is obtained as follows: Figure 5 .

[0156] Adopting such Figure 6 The NARX neural network model with the structure was trained. Given random initial conditions, the model's vibration was solved 100 times. Sampling was performed every 0.001 seconds within 1 second, resulting in 100 sets of input-output data, each with 1001 pairs. Training, validation, and test data were allocated in an 80%, 10%, and 10% ratio, respectively. The neural network was then trained, and a fitting comparison plot was obtained. Figure 6 As shown, the vibration response results of the neural network trained by NARX are very close to those of the real model, indicating that the neural network model fits the real situation well, which confirms the feasibility of applying the NARX model to the establishment of constitutive models of dielectric elastomers.

[0157] like Figure 4 As shown, after obtaining the NARX model, this invention uses a PID control algorithm to control the nonlinear dynamic neural network model of the dielectric elastomer. The proportional parameter is 0.5, the integral parameter is 0.03, and the derivative parameter is 0.1. The feedback quantity is the amplified vibrational expansion and contraction rate of the dielectric elastomer, and the input voltage conforms to a sinusoidal law. Figure 7 The vibration suppression diagram shown demonstrates that the controller can effectively suppress vibration. Figure 8 The tracking desired output effect shown is that after control, the stable vibration frequency, period, and amplitude are the same, but the phase is different. This is mainly determined by the prediction characteristics of the neural network. The phase difference predicted within the range of the sine function is uncontrollable, but the overall control effect is highly complete and can also obtain the desired output with phase difference.

[0158] The parts of this invention not described in detail are well-known to those skilled in the art.

[0159] The above are merely specific steps of the present invention and do not constitute any limitation on the scope of protection of the present invention; it can be extended to the field of data-driven model establishment and active control of intelligent viscoelastic materials. All technical solutions formed by equivalent transformation or equivalent substitution fall within the scope of protection of the present invention.

Claims

1. A neural network-based intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers, used in the design of active control systems that use dielectric elastomers as actuators or sensors, characterized in that... Based on the physical model and the vibration characteristics of the dielectric elastic body, a nonlinear autoregressive neural network with external input is introduced, including the following steps: Step 1: Derive the viscoelastic constitutive model based on Maxwell's rheological model, including: The viscoelasticity of the dielectric elastic body is represented by the Maxwell rheological model, consisting of two nonlinear springs and a damper. The two nonlinear springs are modeled by a hyperelastic model, and the two material parameters are shear modulus, etc. and tensile limit The characteristic of dampers is viscosity. The deformation of an elastic body is described by the stretching in the X and Y directions: and In rheological models The stretching of the spring; The spring's stretch is and The damper's tension in the X and Y directions is and ; The nominal stress tensor is obtained by introducing the strain energy density function. : (1) in, The normal stress is in the X direction. Normal stress in the Y direction; Using a rheological model, the free energy function considering the viscoelasticity of a dielectric elastomer is obtained as follows: (2) in, This represents the actual electric displacement within the elastic body. This refers to the elastic potential energy generated by the deformation of an elastic dielectric in the X direction. This refers to the elastic potential energy generated by the deformation of an elastic dielectric in the Y direction. This represents the electric potential energy of an elastic dielectric during deformation. Step 2: Based on the principle of virtual work, obtain the governing equations for the planar rectangular dielectric elastomer film; Step 3: Derive the nonlinear dynamic differential equations based on the Gent hyperelastic constitutive model, including: Taking dielectric elastomer films as the research object, the initial dimensions of the dielectric elastomer films in the X, Y, and Z directions are lengths of [missing information]. ,width and thickness The dielectric elastomer film is subjected to external forces in the X and Y directions, respectively. and Pre-stretching; when a voltage is applied across the thickness of the elastomer through two flexible electrodes on the top and bottom surfaces of the elastomer, the dielectric elastomer film expands its area in the XY plane and decreases its thickness in the Z direction; the dimensions of the deformed film are expressed as length. ,width and thickness ; The elongation of the film is uniform in the X, Y, and Z directions, and is expressed as follows: , and Assuming the elastic body is incompressible, therefore we have To apply the principle of virtual work, virtual displacement is introduced. and virtual electric displacement Both are smooth in space and independent of time; X and Y are the components of the virtual displacement vector in the X and Y directions, respectively; the virtual work done by the external force P is: (3) in, and It is a defined external force in the in-plane direction, and volume forces are ignored here; The virtual work done by the external voltage is: (4) in, It is the applied voltage. It is the virtual increment of charge on the two electrodes; Let D represent the actual electric displacement in the elastic body, which is related to the total change, as shown below: (5) For uniform biaxial loading, the internal virtual work generated by material deformation is expressed as follows: (6) Where S is the nominal stress tensor and F is the deformation gradient. The initial volume, i.e. ; The internal virtual work generated by the electric field in the material is written as: (7) in, Its own dielectric constant; Virtual inertial work or virtual kinetic energy is: (8) in, Density of the elastic dielectric elastomer; In equilibrium, we have: (9) For uniform biaxial deformation, we have: (10) Will As independent variables, after simplification, the governing equations for the dielectric elastomer film under uniform biaxial deformation are obtained: (11) (12) Step 4: Based on the data obtained from the simulation of the nonlinear mechanics model, a neural network model is constructed using machine learning algorithms. Step 5: Based on the nonlinear active control method, the neural network model is used to achieve the effects of suppressing nonlinear vibration and tracking the desired output of the dielectric elastic body vibration response, including; After obtaining the NARX model, a PID control algorithm is used to control the nonlinear dynamic neural network model of the dielectric elastomer. In the closed-loop system, for the PID controller: The deviation between the given value and the actual output is expressed as: (24) in, For a given value, This is the actual output value of the system; The control law is The transfer function is now: In the formula, This is the proportionality coefficient. The integral time constant is... The differential time constant is These are the Laplace transforms of the output and input, respectively. Debugging is performed according to the control objectives; For the active control structure of the dielectric elastomer parameter vibration system, the active excitation is the applied voltage. The applied voltage is adjusted by the PID controller and fed into the neural network model. The neural network model generates a real-time feedback signal and achieves closed-loop control through PID control. To meet the different needs of suppressing vibration and tracking the desired output, the parameters are adjusted to achieve the control effect.

2. The intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers based on neural networks according to claim 1, characterized in that: The third step includes; The strain energy function of the Gent hyperelastic model is as follows: ； in, It is a volume ratio. These are the corresponding first strain tensor invariants; As the initial shear modulus It is the maximum average deformation parameter. It is an inherent incompressible parameter of the material; Combining incompressible formula And the ideal dielectric elastic body model: ； get: (14) Therefore, the dynamic governing equations are derived based on the constitutive model: (15) (16) (17) (18) in, for The second derivative with respect to time, for The second derivative with respect to time; Set up using the prescribed dynamic model and Where η is viscosity; (19) (20) Consider the case of isoaxial deformation, i.e. , , and At the moment vibration begins, the damper does not have enough time to deform, therefore The damper changes over time; the elastic body The time-varying deformation is calculated from the equilibrium equations with initial conditions: (21) (22) After obtaining the above system of equations, solve the equations.

3. The intelligent active control method for nonlinear dynamics of viscoelastic dielectric elastomers based on neural networks according to claim 2, characterized in that: The fourth step includes: Based on the above model and the numerical solution of the response obtained from simulation, a data-driven free vibration model of a dielectric elastic body is constructed. The neural network used is a NARX nonlinear discrete system model, expressed as: (23) In equation (23): u(t) and y(t) represent the input and output values ​​of the network at time t, respectively; where The maximum order of the input time extension steps; The maximum order of the output time extension steps; therefore All are stored inputs relative to time t; For storage input relative to time t; Let be the nonlinear function obtained by fitting through a neural network; where the subordinate output signal The next value is derived by regression based on the previous values ​​of the output signal and the previous values ​​of the independent input signals; where As input and output values, the hidden layer is set to 3 layers, with 10 neurons in each layer, and is trained. Then, data is prepared for validation to obtain a data-driven neural network model; the neural network is retrained multiple times each time it is trained.

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