Self-adaptive stochastic dynamics analysis method for triboelectric energy collector under high-dimensional excitation
By using a uniform partitioning and adaptive iterative framework based on weighted exponents, the problems of high computational cost and insufficient accuracy in high-dimensional stochastic dynamic analysis are solved, and efficient, accurate stochastic response and reliability assessment of triboelectric energy harvesters are achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- DALIAN UNIV OF TECH
- Filing Date
- 2026-02-03
- Publication Date
- 2026-05-15
AI Technical Summary
Existing high-dimensional stochastic dynamic analysis methods are computationally expensive in triboelectric energy harvesters and lack adaptive mechanisms, resulting in wasted computational resources or loss of accuracy, making it difficult to efficiently and accurately evaluate the system's stochastic response and reliability.
By employing a uniform partitioning technique based on weighted exponents and an adaptive iterative framework, representative points and their assigned probabilities are efficiently determined through the generation of a fixed sample pool and an adaptive iterative process. The probability density function and reliability of the system response are then calculated using the direct probability integral method.
This study achieves efficient and accurate evaluation of triboelectric energy harvesters under high-dimensional random excitation, reduces computational costs, overcomes the problem of uneven distribution of representative points, and provides an efficient framework for evaluating random response and reliability.
Smart Images
Figure CN122046709A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of triboelectric energy harvester technology, and relates to an adaptive stochastic dynamics analysis method for triboelectric energy harvesters under high-dimensional excitation. It is an adaptive stochastic dynamics analysis method for high-dimensional nonlinear stochastic response analysis and reliability assessment of triboelectric energy harvesters. Background Technology
[0002] Triboelectric energy harvesters, as an emerging green energy technology, have broad application prospects in fields such as micro-nano energy, self-powered sensing, and the Internet of Things. Inevitably, triboelectric energy harvesters are affected by environmental excitations such as wind loads and water waves in actual working environments. These external excitations often exhibit significant randomness and broadband characteristics. When using numerical simulation methods such as spectral representation to discretize and model these random excitations, hundreds or even thousands of random phase angle variables are usually introduced to accurately characterize the random characteristics of the excitation. Coupled with the inherent uncertainties in the system's material parameters and geometric dimensions, the system exhibits typical high-dimensional stochastic dynamic characteristics. Therefore, developing efficient and accurate high-dimensional stochastic dynamic analysis methods is crucial for the performance prediction and reliability assessment of triboelectric energy harvesters.
[0003] From a mathematical perspective, such uncertainty quantification problems ultimately boil down to high-dimensional numerical integration problems. Currently, the main methods applicable to high-dimensional numerical integration include Monte Carlo methods (MCS), quasi-Monte Carlo methods (QMCS), and sparse grid methods and their improvements. Monte Carlo methods have convergence rates independent of dimension and good versatility, but their convergence speed is slow. To obtain high-precision probability density functions and reliability assessments, tens or even hundreds of thousands of sample points are often required for physical model calculations. For triboelectric energy harvesting systems involving the solution of complex nonlinear dynamic equations, each sample calculation requires time-consuming time-domain integration, resulting in excessively high computational costs and limiting practical applications. While quasi-Monte Carlo methods and sparse grid methods have faster convergence speeds than Monte Carlo methods under certain conditions, their efficiency depends on the integrand having sufficient smoothness, i.e., requiring the integrand to have bounded mixed partial derivatives, with sparse grid methods having even stricter requirements for smoothness. Therefore, the accuracy and efficiency of these methods in the direct application of high-dimensional stochastic dynamics analysis are limited.
[0004] In recent years, the Direct Probability Integral Method (DPIM), developed based on the principle of probability conservation, has provided an effective approach for stochastic dynamics analysis. This method decouples the structural governing equations from the probability density integral equations, enabling efficient acquisition of the probability density functions and reliability of linear and nonlinear stochastic structural responses, and has shown significant application potential in multiple engineering fields. The core of the DPIM lies in the probability space partitioning technique, namely, how to generate representative points and determine their assigned probabilities.
[0005] Traditional direct probability integration methods typically employ Voronoi cell-based partitioning techniques. This technique generates a large number of sampling points within each Voronoi cell using Monte Carlo simulations to calculate the assigned probability of representative points. The computational cost increases dramatically with dimensionality. When dealing with high-dimensional probability spaces (e.g., hundreds of dimensions), the construction of Voronoi cells and the computational cost of assigning probabilities become extremely high, sometimes even prohibiting implementation due to insufficient memory. More importantly, as the dimensionality increases, the limited number of representative points becomes increasingly concentrated in the central region of the probability space, leading to sparse representative points in the peripheral regions. Furthermore, existing methods lack effective adaptive mechanisms, making it difficult to determine the appropriate number of representative points while maintaining accuracy, often resulting in wasted computational resources or loss of accuracy. Therefore, developing an adaptive analysis method that can overcome the curse of dimensionality and efficiently obtain the probability density function and reliability of high-dimensional nonlinear systems is of great significance. Summary of the Invention
[0006] To address the aforementioned problems, this invention proposes an adaptive stochastic dynamics analysis method for triboelectric energy harvesters under high-dimensional excitation, based on the direct probability integral method and the weighted exponential uniform partitioning method. This method is applicable to the stochastic response analysis and reliability assessment of triboelectric energy harvesters and other nonlinear energy harvesting systems under high-dimensional stochastic excitation, achieving an efficient and accurate unified framework for the high-dimensional stochastic dynamic response and reliability of triboelectric energy harvesters. The main components include: establishing a spectral representation model of the high-dimensional stochastic excitation and a nonlinear dynamic model of the triboelectric energy harvester; employing a weighted exponential-based uniform partitioning technique to efficiently determine representative points and their assigned probabilities by generating a fixed sample pool and constructing an adaptive iterative framework; and using the direct probability integral method to accurately and efficiently calculate the probability density function and dynamic reliability of the system response.
[0007] The technical solution of this invention is as follows: An adaptive stochastic dynamics analysis method for triboelectric energy harvesters under high-dimensional excitation includes the following steps: Step (1) Determine the system parameters of the triboelectric energy harvester (including mass, stiffness, damping, geometric dimensions, material properties and circuit parameters), and use the spectral expression method to simulate the high-dimensional random excitation as a series of harmonic superpositions with random phase angles to establish the nonlinear dynamic model of the triboelectric energy harvester; Step (2) Based on the random variable type set in step (1), for the random phase angle vector used to simulate high-dimensional random excitation, generate a number of random variables at once in the high-dimensional probability space. M A random sample set is used to construct a fixed sample pool; each sample point in the sample pool is mapped to the corresponding range of physical parameter values, forming a sample pool containing... MThe complete set of candidate representative points for the specific values of the group of random variables is used to provide definite input parameters for the nonlinear dynamic model described in step (1); Step (3) Extract samples sequentially from the fixed sample pool. N The sample of 0 is used as the initial representative point set. The probability space is divided into uniform grids using a uniform partitioning method based on weight index, and the assigned probability of the initial representative point set is calculated. At the same time, the initial representative point set is substituted into the nonlinear dynamic equation described in step (1) for time domain solution to obtain the displacement, velocity and voltage time history response. The response data is weighted based on the assigned probability, and the corresponding statistical moments are calculated. Step (4) Add representative points and update the assigned probability of the current representative points; only the newly added representative points in this iteration are substituted into the nonlinear dynamic equation described in step (1) to solve in the time domain, obtain their displacement, velocity and voltage responses, merge with the response data of historical representative points to form the updated response matrix, and combine the response data of the current representative points with the assigned probability to calculate the statistical moments of the system response; Step (5) Calculate the relative error of the response statistical moments between the current iteration and the previous iteration. If the relative error meets the preset convergence tolerance, determine whether the number of convergences has reached the set threshold or the maximum number of iterations.
[0008] Step (6) If step (5) determines that the convergence has not occurred and the current number of representative points does not exceed the maximum number of sample points. M Then, Δ is added sequentially from the fixed sample pool described in step (2). N Add a new sample point to the current representative point set and return to steps (4)-(5) to continue iterating until the convergence condition is met; Step (7) The probability density integral equation is solved numerically, and the adaptive Dirac Delta function smoothing technique is used to obtain the probability density functions of system displacement, velocity and voltage; combined with the first exceedance criterion and the Heaviside function, the dynamic reliability of the triboelectric energy harvester is calculated.
[0009] Specifically, step (3) involves uniform partitioning based on weighted exponents and calculating the assigned probabilities, including: (3.1) Determine the sample space interval based on the probability distribution of the random variable [ a , b ] s ; (3.2) In the hypercube [ a , b ] s The Sobol sequence is used to generate the initial uniformly distributed representative point set θ = [ θ 1, θ 2, …, θ N] represents the number of points. N ; (3.3) By uniformly dividing the sample space [ a , b ] s A subspace is generated in the sample space, and a representative region, i.e., the subspace containing the representative points, is determined; the weight index of the representative region in the one-dimensional sample space is evaluated. w q ; (3.4) Calculate the probability of assigning representative points based on the weight index; (3.5) Use GF-bias to estimate the error, and correct and normalize the assigned probability.
[0010] The beneficial effects of this invention are: This method is applicable to the dynamics of triboelectric energy harvesters under high-dimensional random excitations such as wind loads and environmental vibrations. This method constructs an adaptive iterative framework based on a fixed sample pool and incremental computation, which realizes automatic optimization of sample size and reduces computational cost while ensuring convergence accuracy. By utilizing the adaptive direct probability integral method based on the uniform partitioning technique of weighted exponent, the problem of uneven distribution of representative points in high-dimensional probability space is overcome, and a unified framework for efficient and accurate evaluation of the random response and dynamic reliability of triboelectric energy harvesters under high-dimensional random excitation is developed. Attached Figure Description
[0011] Figure 1 This is a flowchart illustrating the implementation of the adaptive stochastic dynamics analysis method for triboelectric energy harvesters with high-dimensional random excitation, as described in this invention.
[0012] Figure 2 This is a schematic diagram of a triboelectric energy harvester under random excitation provided in an embodiment of the present invention, wherein (a) represents a schematic diagram without collision and (b) represents a schematic diagram with collision.
[0013] Figure 3 The response probability density function curve of the triboelectric energy harvester provided in the embodiment of the present invention, wherein (a) represents displacement. x The probability density function of 2, (b) represents velocity. v The probability density function of 2.
[0014] Figure 4 The graphs of time-varying root mean square voltage and displacement standard deviation provided in the embodiments of the present invention are shown, wherein (a) represents the time-varying root mean square voltage and (b) represents the time-varying displacement standard deviation.
[0015] Figure 5The probability density function curves of the response under different gaps are provided in the embodiments of the present invention, wherein (a) represents the displacement under different gaps. x 2 represents the probability density function, and (b) represents the probability density function of the extreme voltage under different gaps.
[0016] Figure 6 The probability density function curves of the response under different Gaussian white noise intensities are provided in the embodiments of the present invention, wherein (a) represents the displacement under different Gaussian white noise intensities. x The probability density function of 2, (b) represents the velocity under different Gaussian white noise intensities. v The probability density function of 2.
[0017] Figure 7 The probability density function curves under different excitation conditions provided in the embodiments of the present invention, wherein (a) represents the displacement under the action of Gaussian white noise and colored noise of the same intensity. x 2 represents the probability density function, and (b) represents the probability density function of voltage under the action of Gaussian white noise and colored noise of the same intensity.
[0018] Figure 8 The reliability curves for different Gaussian white noise intensities are provided for embodiments of the present invention. Detailed Implementation
[0019] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.
[0020] Figure 1 This is a flowchart illustrating the implementation of an adaptive stochastic dynamics analysis method for a triboelectric energy harvester considering high-dimensional stochastic excitation. It includes the following steps: Step (1): Determine the system parameters of the triboelectric energy harvester (including mass, stiffness, damping, geometric dimensions, material properties and circuit parameters), and use the spectral expression method to simulate the high-dimensional random excitation as a series of harmonic superpositions with random phase angles, and establish the nonlinear dynamic model of the triboelectric energy harvester; The triboelectric energy harvester consists of a two-degree-of-freedom cantilever beam structure and a triboelectric layer. The two-degree-of-freedom cantilever beam structure comprises a main beam and a secondary beam, with the secondary beam consisting of an inertial mass block. M 1. Connection. The secondary beam is connected to the inertial mass block in the opposite direction to the primary beam, where deflection of the primary beam will induce vibration of the secondary beam. The triboelectric layer is attached to the secondary inertial mass block. M The structure consists of an upper aluminum electrode at the bottom and a lower PDMS layer bonded to another aluminum fixed electrode. The upper aluminum electrode and the PDMS layer are separated by an initial gap. g i,2 Separated. When the device vibrates under random excitation, the deflection of the secondary beam will exceed the gap distance, resulting in an impact. Its equation of motion is: (1) in, Represents the equivalent mass of the main beam. For the equivalent mass of the secondary beam, M 1 represents the mass of the inertial mass block. M 2 represents the mass of the secondary inertial mass block. m 1 and m 2 represents the mass of the main beam and the secondary beam, respectively. L 1 and L 2 represents the lengths of the main beam and the secondary beam, respectively. y 1 and y 2 represents the displacement of the main beam and secondary beam of the triboelectric energy harvester, respectively. and The speeds of the main beam and secondary beam of the triboelectric energy harvester. and The speeds of the main beam and secondary beam of the triboelectric energy harvester. M eq1 and M eq2 This represents the equivalent mass of the main beam and the secondary beam. c 11 The damping coefficient of the main beam. c 22 This is the damping coefficient of the secondary beam. c 12 and c 21 This is the coupling damping coefficient. k 11 The stiffness coefficient of the main beam. k 22 This is the stiffness coefficient of the secondary beam. k 12 and k 21 This is the coupling stiffness coefficient. k i,2 and c i,2 For collision stiffness and damping, g i,2 The initial gap distance between the aluminum electrode and the PDMS layer when the triboelectric energy harvester is stationary; electrostatic force. F e,2 Represented as: (2) in, For charge quantity, and The dielectric constants of vacuum and PDMS, S It refers to the contact area. The electromechanical coupling equation is expressed as: (3) in, R It is a resistor. T It is the thickness of the PDMS layer. It is the surface charge density. d o,2 It is the initial gap distance between the aluminum electrodes when the structure is stationary.
[0021] Voltage Represented as: (4) random incentives ξ ( t Modeling is performed using spectral representation methods: , The random phase angle is uniformly distributed in the interval [0, 2π] (which is a set uniformly distributed random variable). Let be the power spectral density function. For frequency increment, For the first k Each frequency point, s This represents the total number of harmonic components.
[0022] Step (2): Based on the random variable type set in step (1), generate a number of random phase angle vectors used to simulate high-dimensional random excitation in the high-dimensional probability space at once. M A random sample set is used to construct a fixed sample pool; each sample point in the sample pool is mapped to the corresponding range of physical parameter values, forming a sample pool containing... M The complete set of candidate representative points for the specific values of the group of random variables is used to provide definite input parameters for the nonlinear dynamic model described in step (1).
[0023] Step (3): Extract samples sequentially from the fixed sample pool. N The sample with 0 is used as the initial representative point set. The probability space is divided into uniform grids using a uniform partitioning method based on weight index, and the assigned probability of the initial representative point set is calculated. At the same time, the initial representative point set is substituted into the nonlinear dynamic equation described in step (1) for time domain solution to obtain the displacement, velocity and voltage time history responses. The response data is weighted based on the assigned probability, and the corresponding statistical moments are calculated.
[0024] (3.1) Determine the sample space interval based on the probability distribution of the random variable [ a , b ] s ; (3.2) In the hypercube [ a , b ] s The Sobol sequence is used to generate the initial uniformly distributed representative point set θ = [ θ1, θ 2, …, θ N ] represents the number of points. N ; (3.3) By uniformly dividing the sample space [ a , b ] s A subspace is generated in the sample space, and a representative region, i.e., the subspace where the representative points are located, is determined. The weight index of the representative region in the one-dimensional sample space is evaluated using equation (5). : (5) in, The total number of representative points, For the volume of the subspace, The total volume of the sample space. For the first q The number of sampling points in each subspace Let be the target probability density function. Let be a uniform probability density function; in the entire probability space, these finite random events are denoted as sample points. and Among them, uniformly distributed random variables u The quantity is set to 10 5 .calculate s The 3D sample space q Weight index of each region : (6) (3.4) Calculate the probability of assigning representative points based on the weight index. : (7) (3.5) Error estimation is performed using the GF-bias, and the assigned probabilities are corrected and normalized. The smaller the GF-bias, the smaller the evaluation error of the regional weight index. GF-bias E GF The calculation formula is: (8) in, s It is the dimension of the probability space. F i (·) indicates the first i The cumulative distribution function of dimension, F e , i (·) represents the empirical cumulative distribution function as follows: (9) in, I {·} indicates an indicator function. For the first q The representative point is at the i The value of dimension, Let represent any point in the probability space. When At that time, determine Representative point of the maximum value Then, the weight index of the representative point is adjusted.
[0025] (10) Among them, the probability is assigned From equation (7), let f U ( θ q,i ) = f U ( θ m Substituting equation (7) into equation (10), we obtain... (11) in, It is the most representative point The weight index, based on equation (11), is the weight index for each dimension. Re-evaluation and substitution into equations (5) and (6) yields the corrected probability of each representative point. .
[0026] Step (4): Add representative points and update the assigned probability of the current representative points; only substitute the newly added representative points in this iteration into the nonlinear dynamic equation described in step (1) to solve in the time domain, obtain their displacement, velocity and voltage responses, merge with the response data of historical representative points to form the updated response matrix, and combine the response data of the current representative points with the assigned probability to calculate the statistical moments of the system response; Based on the direct probability integral method, the time-varying mean of the system response Standard deviation The calculation formula is expressed as: (12) Among them, Ω Θ The sample space representing the input vector, Θ = [Θ1, Θ2, …, Θ n ] indicates that the system input random variable is the random phase angle that the random excitation described in step (1) follows a uniform distribution; Y ( t ) is the response random variable, It is the joint probability density function of Θ. g (θ, t) represents the response value.
[0027] (13) Step (5): Calculate the relative error of the response statistical moments between the current iteration and the previous iteration. If the relative error meets the preset convergence tolerance, determine whether the number of convergences has reached the set threshold or the maximum number of iterations.
[0028] Calculate the current iteration k Previous iteration k -1 relative error of the response standard deviation : (14) Determine whether the convergence criterion is met. ≤ ε 0, where This is the preset convergence threshold; If the convergence criterion is met, then the number of consecutive convergences is... N std Increment the count by 1; like N std ≥ N tol Or reach the maximum number of iterations iter max If the iteration stops, then stop. If the stopping condition is not met and the current point is... N current ≤ M Then, the sample from the fixed sample pool is taken (the previous sample). N current +Δ N ) points are used as the new representative point set, where Δ N This represents the number of points added each time.
[0029] Step (6): If step (5) determines that the convergence has not occurred and the current number of representative points does not exceed the maximum number of sample points. M Then, Δ is added sequentially from the fixed sample pool described in step (2). N Add a new sample point to the current representative point set and return to steps (4)-(5) to continue iterating until the convergence condition is met; Step (7): By numerically solving the probability density integral equation, the probability density functions of system displacement, velocity and voltage are obtained using the adaptive Dirac-delta function smoothing method; the dynamic reliability of the triboelectric energy harvester is calculated by combining the first exceedance criterion and the Heaviside function.
[0030] Based on the principle of probability conservation, the integral equation of probability density is obtained: (15) in, It is the joint probability density function of the multidimensional random response Y, where y is the vector of specific state values of the random response Y. Let be the Dirac function, random variable Θ, and response Y( t Mapping between ) Y( t )=g(Θ, t ).
[0031] An adaptive uniform partitioning method based on weighted exponents and Dirac δ The adaptive smoothing method for the function is used to solve equation (15) to obtain the probability density function of the response: (16) in, N Represents the total number of sample points; θ q It is the first in the input probability space q One sample point; yes q The probability of assigning a sample point; the smoothing parameter of the Dirac function. It is obtained from the kernel density estimation.
[0032] Simultaneously, the first-breakthrough dynamic reliability is obtained by combining the extreme response, expressed as: (17) in, Represents the function. For the Heaviside function.
[0033] Reference Example 1: Stochastic dynamics analysis of a triboelectric energy harvester under Gaussian white noise (see schematic diagram) Figure 2 Taking the optimized design of the main beam as an example, the dimensions of the main beam are 95×20×1mm, the dimensions of the secondary beam are 135×20×1mm, and the dimensions of the PDMS layer are 40×20×0.4mm. M 1 and M 2 represents the equivalent mass of the main beam and the secondary beam, respectively, which are 0.007 kg and 0.003 kg. M eq1 and M eq2 The equivalent mass of the main beam and secondary beams, c 11 =0.25 Ns / m is the damping coefficient of the main beam. c 22 =0.04Ns / m is the damping coefficient of the secondary beam. c 12 =0.1Ns / m and c 21=0.003Ns / m is the coupling damping coefficient. k 11 =519.91 N / m is the stiffness coefficient of the main beam. k 22 =91.78 N / m is the stiffness coefficient of the secondary beam. k 12 and k 21 =103.86 N / m is the coupling stiffness coefficient; k i,2 =1.19×10 4 N / m and c i,2 =5.2 Ns / m represents the impact stiffness and damping. Resistance R The initial gap distance between the aluminum electrode and the PDMS layer when the structure is stationary is 10 MΩ. g i,2 =1.6mm. This problem utilizes 500 uniformly distributed random variables for the generation of Gaussian white noise.
[0034] The probability density functions of displacement and voltage of the triboelectric energy harvester obtained based on this invention are as follows: Figure 3 As shown, the time-varying root mean square voltage and displacement standard deviation are as follows: Figure 4 As shown in the figure, the Gaussian white noise intensity is 0.01. The calculation results of this invention agree well with the Monte Carlo simulation solution, demonstrating the accuracy of this invention in calculating the probability density function of the random response of the triboelectric energy harvester. Furthermore, the CPU runtime required for the random response analysis of the triboelectric energy harvester is shown in Table 1, indicating that this invention can improve computational efficiency by more than 100 times compared to the Monte Carlo simulation method while ensuring the accuracy of the random response analysis.
[0035] Table 1 Comparison of different methods for solving random responses
[0036] The probability density curves of the triboelectric energy harvester response under different gap conditions are as follows: Figure 5 As shown, as the gap decreases, the effect on displacement is small, but the mode of the extreme voltage increases. This is because the reduced gap induces collisions, thereby increasing the voltage output.
[0037] The probability density function curves of the response under different Gaussian white noise intensities are derived from... Figure 7 The peak probability density of the response decreases as the Gaussian white noise intensity increases. The probability density function curves for colored noise and Gaussian white noise of the same intensity are shown below. Figure 7As shown, the system under colored noise excitation exhibits a response suppression effect, while Gaussian white noise has a wider response range and a lower peak voltage compared to colored noise. Reliability curves for different Gaussian white noise intensities are shown below. Figure 8 As shown, reliability decreases with increasing noise intensity.
[0038] In summary, this invention presents an adaptive stochastic dynamics analysis method for triboelectric energy harvesters considering high-dimensional stochastic excitation. It proposes a high-dimensional stochastic dynamics analysis scheme based on a weighted exponential uniform partitioning technique, and utilizes a fixed large-capacity sample pool and an adaptive iterative framework to efficiently and accurately solve for the probability density function of the system's nonlinear response and dynamic reliability. This invention achieves accurate dynamic evaluation of triboelectric energy harvesters under high-dimensional stochastic excitation, fully considering the strong nonlinearity of multi-physics coupled systems and the high-dimensional randomness of input excitation. It significantly overcomes the efficiency bottleneck of high-dimensional numerical computation and develops a unified framework for efficient and accurate evaluation of the stochastic response and dynamic reliability of triboelectric energy harvesters under high-dimensional stochastic excitation.
Claims
1. An adaptive stochastic dynamics analysis method for triboelectric energy harvesters under high-dimensional excitation, characterized in that, Includes the following steps: Step (1) Determine the system parameters of the triboelectric energy harvester, including mass, stiffness, damping, geometric dimensions, material properties and circuit parameters. Use the spectral expression method to simulate the high-dimensional random excitation as a series of harmonic superpositions with random phase angles, and establish the nonlinear dynamic model of the triboelectric energy harvester. Step (2) Based on the random variable type set in step (1), for the random phase angle vector used to simulate high-dimensional random excitation, generate a number of random variables at once in the high-dimensional probability space. M A random sample set is used to construct a fixed sample pool; each sample point in the sample pool is mapped to the corresponding range of physical parameter values, forming a sample pool containing... M The complete set of candidate representative points for the specific values of the group of random variables is used to provide definite input parameters for the nonlinear dynamic model described in step (1); Step (3) Extract samples sequentially from the fixed sample pool. N The sample of 0 is used as the initial representative point set. The probability space is divided into uniform grids using a uniform partitioning method based on weight index, and the assigned probability of the initial representative point set is calculated. At the same time, the initial representative point set is substituted into the nonlinear dynamic equation described in step (1) for time domain solution to obtain the displacement, velocity and voltage time history response. The response data is weighted based on the assigned probability, and the corresponding statistical moments are calculated. Step (4) Add representative points and update the assigned probability of the current representative points; only the newly added representative points in this iteration are substituted into the nonlinear dynamic equation described in step (1) to solve in the time domain, obtain their displacement, velocity and voltage responses, merge with the response data of historical representative points to form the updated response matrix, and combine the response data of the current representative points with the assigned probability to calculate the statistical moments of the system response; Step (5) Calculate the relative error of the response statistical moments between the current iteration and the previous iteration. If the relative error meets the preset convergence tolerance, determine whether the number of convergences has reached the set threshold or the maximum number of iterations. Step (6) If step (5) determines that the convergence has not occurred and the current number of representative points does not exceed the maximum number of sample points. M Then, Δ is added sequentially from the fixed sample pool described in step (2). N Add a new sample point to the current representative point set and return to steps (4)-(5) to continue iterating until the convergence condition is met; Step (7) The probability density integral equation is solved numerically, and the adaptive Dirac Delta function smoothing technique is used to obtain the probability density functions of system displacement, velocity and voltage; combined with the first exceedance criterion and the Heaviside function, the dynamic reliability of the triboelectric energy harvester is calculated.
2. The adaptive stochastic dynamics analysis method for a triboelectric energy harvester under high-dimensional excitation according to claim 1, characterized in that, Step (1) is as follows: The triboelectric energy harvester consists of a two-degree-of-freedom cantilever beam structure and a triboelectric layer; the two-degree-of-freedom cantilever beam structure comprises a main beam and a secondary beam, the secondary beam being composed of an inertial mass block. M 1. Connection; the secondary beam is connected to the inertial mass block in the opposite direction to the primary beam, wherein the deflection of the primary beam will induce the vibration of the secondary beam; the triboelectric layer is attached to the secondary inertial mass block. M 2. The upper aluminum electrode at the bottom and the lower PDMS layer bonded to another aluminum fixed electrode are composed of an upper aluminum electrode and a lower PDMS layer; the upper aluminum electrode and the PDMS layer are separated by an initial gap. g i,2 Separate; When the device vibrates under random excitation, the deflection of the secondary beam will exceed the gap distance, resulting in an impact. Its equation of motion is: (1) ； in, Represents the equivalent mass of the main beam. For the equivalent mass of the secondary beam, M 1 represents the mass of the inertial mass block. M 2 represents the mass of the secondary inertial mass block. m 1 and m 2 represents the mass of the main beam and the secondary beam, respectively. L 1 and L 2 represents the lengths of the main beam and the secondary beam, respectively. y 1 and y 2 represents the displacement of the main beam and secondary beam of the triboelectric energy harvester, respectively. and The speeds of the main beam and secondary beam of the triboelectric energy harvester. and The speeds of the main beam and secondary beam of the triboelectric energy harvester. M eq1 and M eq2 This represents the equivalent mass of the main beam and the secondary beam. c 11 The damping coefficient of the main beam. c 22 This is the damping coefficient of the secondary beam. c 12 and c 21 This is the coupling damping coefficient. k 11 The stiffness coefficient of the main beam. k 22 This is the stiffness coefficient of the secondary beam. k 12 and k 21 This is the coupling stiffness coefficient. k i,2 and c i,2 For collision stiffness and damping, g i,2 The initial gap distance between the aluminum electrode and the PDMS layer when the triboelectric energy harvester is stationary; electrostatic force. F e,2 Represented as: (2) ； in, For charge quantity, and The dielectric constants of vacuum and PDMS, S It is the contact area; the electromechanical coupling equation is expressed as: (3) ； in, R It is a resistor. T It is the thickness of the PDMS layer. It is the surface charge density. d o,2 It is the initial gap distance between the aluminum electrodes when the structure is stationary; Voltage Represented as: (4) ； random incentives ξ ( t Modeling is performed using spectral representation: , Let be a random phase angle uniformly distributed in the interval [0, 2π], and be a uniformly distributed random variable. Let be the power spectral density function. For frequency increment, For the first k Each frequency point, s This represents the total number of harmonic components.
3. The adaptive stochastic dynamics analysis method for a triboelectric energy harvester under high-dimensional excitation according to claim 1, characterized in that, Step (3) specifically includes the uniform partitioning based on the weighted index and the calculation of the assigned probability: (3.1) Determine the sample space interval based on the probability distribution of the random variable [ a , b ] s ; (3.2) In the hypercube [ a , b ] s The Sobol sequence is used to generate the initial uniform distribution representative point set θ = [ θ 1, θ 2, …, θ N ] represents the number of points. N ; (3.3) By uniformly dividing the sample space [ a , b ] s A subspace is generated in the sample space, and a representative region, i.e., the subspace containing the representative points, is determined; the weight index of the representative region in the one-dimensional sample space is evaluated. w q ; (3.4) Calculate the probability of assigning representative points based on the weight index; (3.5) Use GF-bias to estimate the error, and correct and normalize the assigned probability.
4. The adaptive stochastic dynamics analysis method for a triboelectric energy harvester under high-dimensional excitation according to claim 3, characterized in that, Step (3) is as follows: (3.1) Determine the sample space interval based on the probability distribution of the random variable [ a , b ] s ; (3.2) In the hypercube [ a , b ] s The Sobol sequence is used to generate the initial uniform distribution representative point set θ = [ θ 1, θ 2, …, θ N ] represents the number of points. N ; (3.3) By uniformly dividing the sample space [ a , b ] s A subspace is generated within the subspace, and a representative region is determined, that is, the subspace where the representative point is located. The weight index of representative regions in the one-dimensional sample space is evaluated using equation (5). : (5) ； in, The total number of representative points, For the volume of the subspace, The total volume of the sample space. For the first q The number of sampling points in each subspace Let be the target probability density function. Let be a uniform probability density function; in the entire probability space, these finite random events are denoted as sample points. and ;calculate s The 3D sample space q Weight index of each region : (6) ； (3.4) Calculate the probability of assigning representative points based on the weight index. : (7) ； (3.5) Error estimation is performed using the GF-bias, and the assigned probabilities are corrected and normalized; the smaller the GF-bias, the smaller the evaluation error of the regional weight index; GF-bias E GF The calculation formula is: (8) ； in, s It is the dimension of the probability space. F i (·) indicates the first i The cumulative distribution function of dimension, F e , i (·) represents the empirical cumulative distribution function as follows: (9) ； in, I {·} indicates an indicator function. For the first q The representative point is at the i The value of dimension, Represent any point in the probability space; when At that time, determine The representative point of the maximum value Then, adjust the weight index of the representative point; (10) ； Among them, the probability is assigned From equation (7), let f U ( θ q,i ) = f U ( θ m Substituting equation (7) into equation (10), we obtain... (11) ； in, It is the most representative point The weight index, based on equation (11), is the weight index for each dimension. Re-evaluation and substitution into equations (5) and (6) yields the corrected probability of each representative point. .
5. The adaptive stochastic dynamics analysis method for a triboelectric energy harvester under high-dimensional excitation according to claim 1, characterized in that, Step (4) is as follows: Based on the direct probability integral method, the time-varying mean of the system response Standard deviation The calculation formula is expressed as: (12) ； Among them, Ω Θ The sample space representing the input vector, Θ = [Θ1, Θ2, …, Θ n ] indicates that the system input random variable is the random phase angle that the random excitation described in step (1) follows a uniform distribution; Y ( t ) is the response random variable, It is the joint probability density function of Θ. g (θ, t () represents the system response value; (13)。 6. The adaptive stochastic dynamics analysis method for a triboelectric energy harvester under high-dimensional excitation according to claim 1, characterized in that, Step (5) is as follows: Calculate the current iteration k Previous iteration k -1 relative error of the response standard deviation : (14) ； Determine whether the convergence criterion is met. ≤ ε 0, where This is the preset convergence threshold; If the convergence criterion is met, then the number of consecutive convergences is... N std Increment the count by 1; like N std ≥ N tol Or reach the maximum number of iterations iter max If the iteration stops, then stop. If the stopping condition is not met and the current point is... N current ≤ M Then, the sample from the fixed sample pool is taken (the previous sample). N current +Δ N ) points are used as the new representative point set, where Δ N This represents the number of points added each time.
7. The adaptive stochastic dynamics analysis method for a triboelectric energy harvester under high-dimensional excitation according to claim 1, characterized in that, Step (7) is as follows: Based on the principle of probability conservation, the integral equation of probability density is obtained: (15) ； in, It is the joint probability density function of the multidimensional random response Y, where y is the vector of specific state values of the random response Y; Let be the Dirac function, random variable Θ, and response Y( t Mapping between ) Y( t )=g(Θ, t ); An adaptive uniform partitioning method based on weighted exponents and Dirac δ The adaptive smoothing method for the function is used to solve equation (15) to obtain the probability density function of the response: (16) ； in, N Represents the total number of sample points; θ q It is the first in the input probability space q One sample point; yes q The probability of assigning a sample point; the smoothing parameter of the Dirac function. Obtained from kernel density estimation; Simultaneously, the first-breakthrough dynamic reliability is obtained by combining the extreme response, expressed as: (17) ； in, Represents the function, For the Heaviside function.