Reliability evaluation system and method for two-degree-of-freedom rotational vibration energy collector

Abstract

The invention provides a reliability evaluation system and method for a two-degree-of-freedom rotational vibration energy collector, and belongs to the field of nonlinear rotational vibration energy collection, and the method comprises the steps: S1, constructing a kinetic model, and converting the kinetic model into a Hamiltonian form; s2, reducing dimensions by adopting a random average method; s3, calculating a drift coefficient and a diffusion coefficient; s4, constructing and solving a boundary value problem to verify the calculation precision; and S5, parameter sensitivity analysis and design optimization. According to the reliability evaluation system and method for the two-degree-of-freedom rotary vibration energy collector, the problems that in reliability analysis of an existing two-degree-of-freedom rotary vibration energy collector, high-dimensional coupling characteristics are difficult to capture, an analysis frame is incomplete, calculation efficiency is low, and parameter optimization lacks theoretical guidance are solved.

Description

A reliability assessment system and method for a two-degree-of-freedom rotational vibration energy harvester Technical Field

[0001] This invention relates to the field of nonlinear rotational vibration energy harvesting, and in particular to a reliability assessment system and method for a two-degree-of-freedom rotational vibration energy harvester. Background Technology

[0002] Rotary vibration energy harvesters, as effective devices for converting mechanical energy in the environment into electrical energy, demonstrate significant self-powered application value in low-power systems such as the Internet of Things, smart tire sensors, and embedded devices in spacecraft. Current research on rotary vibration energy harvesters primarily focuses on their structural design and performance optimization, covering key areas such as fundamental dynamic analysis, improving mechanical energy capture efficiency, and enhancing adaptability to various operating conditions.

[0003] Traditionally, the design of rotating vibration energy harvesters is mostly based on linear models. These models have simple structures but narrow effective bandwidths, exhibiting efficient energy conversion characteristics only near the resonant frequency, making them unsuitable for the complex conditions of wide-frequency amplitude variations in real-world vibration environments. Therefore, nonlinear design has gradually become a research hotspot, with bistable, aeroelastic, and double-beam coupled nonlinear structures being proposed to broaden the operating bandwidth and improve energy harvesting capabilities through nonlinear mechanisms. Among these, nonlinear rotating vibration energy harvesters have shown superior performance to linear systems under specific conditions, attracting widespread attention.

[0004] However, nonlinear rotating vibration energy harvesters face challenges in practical operation due to the coupling effect of structural nonlinearity and random excitation. Their reliability and failure mechanisms have not been fully quantified, becoming a key bottleneck restricting their engineering applications. Although nonlinear design can effectively broaden the bandwidth, the reliability problem of the system remains prominent under random excitation. The coupling of deterministic nonlinearity and random disturbances can easily induce the system response to exceed the safety threshold, leading to functional failure or structural damage.

[0005] First-break time analysis, as a core tool in the reliability study of nonlinear stochastic systems, aims to characterize the statistical time characteristics of the system response when it first crosses the safety threshold. Its core indicators include the conditional reliability function, the first-break time probability density function, and the mean first-break time, providing quantitative support for structural life prediction and safety design. However, existing research mostly focuses on general nonlinear systems, single-degree-of-freedom systems, or chain structures. In-depth research on the dynamic reliability analysis framework, failure mechanism analysis, and first-break time behavior of two-degree-of-freedom nonlinear rotating vibration energy harvesters, which possess high-dimensional dynamic characteristics, strong nonlinearity, and stochastic excitation coupling effects, remains relatively scarce.

[0006] Furthermore, real-world systems exhibit significant nonlinear characteristics due to factors such as structure and friction, which severely limits the prediction accuracy of traditional linear models. Although existing studies have explored reliability modeling and first-pass problem solutions for systems under nonlinear or switching excitation using methods such as stochastic averaging, Markov jump processes, and two-step generalized elliptic coordinate transformation, specific research on two-degree-of-freedom rotating vibration energy harvesters remains insufficient. Summary of the Invention

[0007] The purpose of this invention is to provide a reliability assessment system and method for a two-degree-of-freedom rotating vibration energy harvester, which solves the problems in the existing reliability analysis of two-degree-of-freedom rotating vibration energy harvesters, such as difficulty in capturing high-dimensional coupling characteristics, incomplete analysis framework, low computational efficiency, and lack of theoretical guidance for parameter optimization.

[0008] To achieve the above objectives, this invention provides a reliability assessment method for a two-degree-of-freedom rotating vibration energy harvester, comprising the following steps: S1, Dynamic Modeling: For a two-degree-of-freedom rotating vibration energy harvester containing a master oscillator and a rotor branch, a nonlinear dynamic model considering Coulomb friction terms is established, system parameters are defined, and the equations of motion of the dynamic model are transformed into Hamiltonian form through variable transformation; S2, Stochastic Averaging Dimensionality Reduction: The quasi-nonintegrable Hamiltonian system obtained in step S1 is subjected to stochastic averaging, and the rapidly changing variables are averaged over time to simplify the high-dimensional coupled system into a one-dimensional stochastic energy diffusion process containing only total energy, so as to retain the key characteristics of the system's energy evolution; S3, Drift and Diffusion Coefficient Calculation: Based on the one-dimensional energy diffusion process described in step S2, the drift coefficient, diffusion coefficient, and energy period of the energy diffusion process are calculated through periodic integration and numerical integration in the energy domain. When a closed-form analytical solution is unavailable, high-precision numerical integration is used for estimation; S4, Reliability Indicators Solution and Verification: Using the energy threshold as the failure criterion, backward Kolmogorov equations and generalized Pontryagin equations are constructed in the energy domain. Boundary value problems are solved by combining the corresponding initial and boundary conditions to obtain the conditional reliability function, the conditional probability density function (CPDF) of the first crossing time, and the mean first crossing time (MFPT). The calculation results of the conditional reliability function, CPDF, and MFPT are compared and verified using Monte Carlo simulation. S5. Parameter Sensitivity Analysis and Design Optimization: Based on the reliability indices obtained in step S4, a parameter set including initial energy, noise intensity, natural frequency, mechanical damping ratio, electromagnetic damping ratio, and Coulomb friction coefficient is selected. A perturbation of a predetermined amplitude is applied to each parameter in the parameter set. Instantaneous elasticity indices are defined and calculated to obtain the average elasticity indices at multiple representative moments, thereby quantifying the sensitivity level of each parameter and forming an engineering adjustment and optimization scheme for reliability design.

[0009] Preferably, the dynamic model includes the main oscillator and the rotor branch, and the equation of motion is expressed as follows: ; ;in, For quality, The mechanical damping coefficient is... The electromagnetic damping coefficient is... , All are linear stiffness coefficients. The stiffness coefficient is a third-order nonlinear coefficient. For quality The relative displacement between the base and the support. For generator The relative displacement between the base and the support. It is the inertial constant. Let be the value of the Coulomb friction force. It is the acceleration of the base; the Hamiltonian equation of motion is: ; ;in, , , , , and For the damping ratio, , , It is the system frequency. is the Coulomb friction coefficient.

[0010] Preferably, the stochastic averaging method is used to average the rapidly changing variables over time, eliminating the high-frequency oscillations: The total energy is defined as follows: ;in, The energy of the main oscillator The energy is in the rotor branch; the expression for the one-dimensional energy diffusion process is: ;in, The drift coefficient, Where is the diffusion coefficient. This is standard Brownian motion.

[0011] Preferably, the expressions for the drift coefficient and the diffusion coefficient are as follows: ; ;in, For energy cycles, The domain of integration is defined as follows: The energy period expression is: The expression for the domain of the integral is: .

[0012] Preferably, the drift coefficient, diffusion coefficient, and energy period of the energy diffusion process are given by energy domain integration, and high-precision numerical integration estimation is used when the closed-form solution is unavailable.

[0013] Preferably, the failure criterion sets the upper boundary of the energy domain as an absorbing boundary and the lower boundary as a reflecting or natural boundary; the equation expression for the backward Kolmogorov (BK) is: The equation for the generalized Pontryagin (PG) is: The conditional reliability function expression is: The CPDF expression for the first time travel is: The MFPT expression is: ;MFPT satisfies the PG equation: ;MFPT is given by mutual proof of the BK equation and the PG equation.

[0014] Preferably, the parameter sensitivity analysis includes: selecting a set of sensitive parameters covering initial energy, noise intensity, natural frequency, nonlinear stiffness coefficient, mechanical damping ratio, electromagnetic damping ratio, and Coulomb friction intensity; and a quantification method by applying a 20% perturbation to each parameter in the parameter set, defining an instantaneous elastic index, taking the average elastic index at multiple times, and ranking the parameter sensitivity levels.

[0015] Preferably, the expression for the instantaneous elasticity index is: ;in, For parameters, The reference value for the parameter is... The reliability function for the parameters. This is a reliability function for the parameter baseline value.

[0016] Preferably, engineering parameter tuning recommendations include: providing an initial energy safety margin control strategy for the installation and startup phases; and providing a vibration isolation / damping and lubrication collaborative design strategy for random strong excitation scenarios.

[0017] A reliability assessment system for a two-degree-of-freedom rotating vibration energy harvester includes: a processor and a memory, wherein the memory stores a computer program executable on the processor; wherein, when executing the computer program, the processor is configured as a dynamic modeling module for establishing a nonlinear dynamic model of the two-degree-of-freedom rotating vibration energy harvester, incorporating Coulomb friction terms, and transforming the equations of motion of the dynamic model into Hamiltonian form; and a stochastic averaging dimensionality reduction module for applying a stochastic averaging method to the quasi-nonintegrable system of Hamiltonian form, reducing the high-dimensional coupled system to a one-dimensional energy diffusion process containing only total energy, and evaluating the drift coefficient and diffusion coefficient of the energy diffusion process. The system includes a first-crossing index (FCI) calculation module, which constructs and solves the boundary value problems of the backward Kolmogorov equation and the generalized Pontryagin equation in the energy domain to obtain the conditional reliability function, the CPDF of the first-crossing time, and the MFPT. The calculation results can be compared and verified with Monte Carlo simulation results. The parameter sensitivity analysis and optimization module calculates the average elastic sensitivity index based on the conditional reliability function and the first-crossing time statistics. It ranks the sensitivity of initial energy, noise intensity, natural frequency, mechanical damping ratio, electromagnetic damping ratio, and Coulomb friction parameters, and outputs reliability-oriented engineering adjustment and design optimization schemes.

[0018] Therefore, the present invention adopts the above-mentioned reliability assessment system and method for a two-degree-of-freedom rotating vibration energy harvester, and the technical effects are as follows: 1. Efficient dimensionality reduction and simplified calculation: For complex systems with high dimensions, strong nonlinearity and coupling damping, the stochastic averaging method (SAM) is used to reduce the system to a one-dimensional energy diffusion process, which significantly simplifies the calculation process and avoids the complexity and computational burden brought about by direct analysis of high-dimensional systems; the drift and diffusion coefficients are obtained by numerical integration in the energy domain, which further improves the computational efficiency, making the reliability analysis of complex nonlinear systems feasible and efficient.

[0019] 2. Accurate assessment of reliability indicators: Using energy threshold as the failure criterion, backward Kolmogorov (BK) and generalized Pontryagin (GP) boundary value problems are constructed and solved, resulting in conditional reliability function, first pass time CPDF and MFPT. This provides quantitative support for system lifetime prediction and safety design, and helps to accurately assess the reliability level of the system under different operating conditions.

[0020] 3. Quantitative Parameter Influence and Sensitivity Analysis: The influence of initial energy, noise intensity, natural frequency, and damping parameters on system reliability is quantified, revealing the modulation effect of each parameter on system reliability. Parameter sensitivity analysis is conducted using the average elasticity index, clarifying the hierarchy of parameter influence on system reliability and providing a quantitative basis for parameter prioritization in reliability-oriented design. Attached Figure Description

[0021] Figure 1 is a schematic diagram of the two-degree-of-freedom rotational vibration energy harvester of the present invention; Figure 2 is a drift diffusion coefficient diagram of an embodiment of the present invention; (a) is The surface drift coefficient and diffusion coefficient; (b) is The surface drift coefficient and diffusion coefficient; (c) is The surface drift coefficient and diffusion coefficient; (d) is The surface drift coefficient and diffusion coefficient; (e) is The surface drift coefficient and diffusion coefficient; (f) is The drift coefficient and diffusion coefficient of the surface; Figure 3 is a cross-sectional view of the drift diffusion coefficient of an embodiment of the present invention; (a) is (a) Cross-sectional view of the surface drift coefficient and diffusion coefficient; (b) is Cross-sectional views of the surface drift coefficient and diffusion coefficient; (c) is... Cross-sectional views of the surface drift coefficient and diffusion coefficient; (d) is... Cross-sectional views of the surface drift coefficient and diffusion coefficient; (e) is... Cross-sectional views of the surface drift coefficient and diffusion coefficient; (f) is... Figure 4 shows a cross-sectional view of the drift coefficient and diffusion coefficient of the surface; Figure 4 is a comparison and verification diagram of the reliability function of the Monte Carlo simulation method and the stochastic averaging method in the embodiment of the present invention; (a) is three-dimensional; (b) is two-dimensional planar ( Figure 5 shows a comparison of the first-pass PDF between the Monte Carlo simulation method and the stochastic averaging method according to an embodiment of the present invention; (a) is three-dimensional; (b) is two-dimensional planar. Figure 6 shows a comparison and verification diagram of the Monte Carlo simulation method and the stochastic averaging method using MFPT in an embodiment of the present invention; (a) is three-dimensional; (b) is two-dimensional planar; Figure 7 shows different initial energy in an embodiment of the present invention. Schematic diagram of the impact on system conditional reliability function and first crossover time PDF; (a) system conditional reliability function; (b) first crossover time PDF; Figure 8 shows different noise intensities in the embodiments of the present invention. Schematic diagram of the impact on system conditional reliability function, first cross-time PDF, and MFPT; (a) system conditional reliability function; (b) first cross-time PDF; (c) MFPT; Figure 9 shows the natural frequencies of different master systems in the embodiments of the present invention. Schematic diagram of the impact on system conditional reliability function, first cross-time PDF, and MFPT; (a) system conditional reliability function; (b) first cross-time PDF; (c) MFPT; Figure 10 shows the natural frequencies of different master systems in the embodiments of the present invention. Schematic diagram of the impact on system conditional reliability function, first cross-time PDF, and MFPT; (a) system conditional reliability function; (b) first cross-time PDF; (c) MFPT; Figure 11 shows the natural frequencies of different master systems in the embodiments of the present invention. Schematic diagram of the impact on system conditional reliability function, first crossing time PDF, and MFPT; (a) system conditional reliability function; (b) first crossing time PDF; (c) MFPT; Figure 12 shows different mechanical damping ratios in embodiments of the present invention. Schematic diagram of the impact on system conditional reliability function, first crossing time PDF, and MFPT; (a) system conditional reliability function; (b) first crossing time PDF; (c) MFPT; Figure 13 shows different electromagnetic damping ratios in embodiments of the present invention. Schematic diagram of the impact on system conditional reliability function, first crossing time PDF, and MFPT; (a) system conditional reliability function; (b) first crossing time PDF; (c) MFPT; Figure 14 shows different Coulomb frictions in embodiments of the present invention. Schematic diagram of the impact on the system conditional reliability function, first cross-time PDF, and MFPT; (a) is the system conditional reliability function; (b) is the first cross-time PDF; (c) is the MFPT. Detailed Implementation

[0022] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0023] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0024] Example 1: This invention provides a reliability assessment method for a two-degree-of-freedom rotating vibration energy harvester, comprising the following steps: S1, constructing a dynamic model of the two-degree-of-freedom rotating vibration energy harvester including the main oscillator and rotor branches, defining system parameters, and incorporating Coulomb friction terms; transforming the equations of motion into Hamiltonian form through variable transformation; S2, for the quasi-non-integrable Hamiltonian system established in S1, using the stochastic averaging method to eliminate rapidly changing variables, simplifying the high-dimensional coupled system into a one-dimensional diffusion process containing only total energy, while retaining the key characteristics of system energy evolution; S3, based on the dimensionality-reduced energy diffusion process, through... S4. Using the energy threshold as the failure criterion, construct the BK equation and PG equation. By solving the boundary value problem, obtain the conditional reliability function, first-crossing time CPDF, and MFPT. Verify the calculation accuracy of the conditional reliability function, first-crossing time CPDF, and MFPT through Monte Carlo simulation. S5. Conduct parameter sensitivity analysis and design optimization based on reliability indicators. Quantify parameter sensitivity through the average elasticity index and output engineering parameter tuning suggestions for initial energy, noise intensity, natural frequency, and damping parameters.

[0025] Dynamics Modeling Module: Establishes a nonlinear dynamic model of a two-degree-of-freedom rotating vibration energy harvester and forms a Hamiltonian representation; Stochastic Averaging Dimensionality Reduction Module: Reduces the system dimension to a one-dimensional energy diffusion process and evaluates drift, diffusion, and energy periodicity; First-Crossover Calculation Module: Constructs and solves the boundary value problems of the BK and PG equations in the energy domain to obtain the conditional reliability function, the CPDF of the first-crossover time, and the MFPT; Parameter Sensitivity and Optimization Module: Conducts parameter sensitivity analysis and formulates engineering adjustment and reliability-oriented design schemes.

[0026] As shown in Figure 1, the two-degree-of-freedom rotational vibration energy harvester consists of linear stiffness... and Nonlinear stiffness ,quality Mechanical damping and rotating parts The components, and their relative displacements relative to the base are respectively and Rotating part It acts as a generator, converting mechanical energy into electrical energy. It can be viewed as an inertial device. and resistance damping Parallel combinations, where forces .

[0027] According to Newton's laws, the equation of motion of the data logger is as follows: ; ;in, For quality, The mechanical damping coefficient is... The electromagnetic damping coefficient is... , All are linear stiffness coefficients. The stiffness coefficient is a third-order nonlinear coefficient. For quality The relative displacement between the base and the support. For generator The relative displacement between the base and the support. It is the inertial constant. Let be the value of the Coulomb friction force. It is the acceleration of the base; using the mean square velocity of the system. The average power is as follows: By introducing the following transformation: We can obtain: ; ;in, and The damping ratio; , , It is the system frequency; Dielectric constant With quality The ratio; The coefficient of friction is Coulomb; while Gaussian white noise It is the cardinality of random stimulus, where the average value is... With Gaussian white noise Related functions At this point, the average power can be rewritten as... .

[0028] When nonlinear stiffness and friction are ignored, the system simplifies to a linear vibration energy harvesting system. This simplified model has been widely used in early first-pass failure studies; however, it cannot capture energy dissipation deviations and response distortions caused by nonlinear factors in actual structures, hindering accurate prediction of the critical failure state. In contrast, this system is a nonlinear vibration energy harvesting system, whose energy evolution exhibits strong nonlinearity. The system's first-pass behavior is coupled with stiffness nonlinearity and nonsmooth damping, significantly increasing the difficulty of calculating the failure probability. Subsequently, the energy domain stochastic evolution equation is derived using the nonlinear model, providing a basis for the quantitative analysis of first-pass failure.

[0029] For nonlinear systems, to avoid the interference of scale mismatch with first-crossing statistics and to obtain a baseline description of the convection-diffusion structure, we consider the ratio. In this case, the equation of motion can be expressed as: ; To obtain an analytical solution, a stochastic average of a quasi-non-integrable Hamiltonian system is used. Here, we let... , , , The Hamiltonian form of the equations of motion is: ; ; ; Considering the non-integrability of Hamiltonian systems, a stochastic averaging method was adopted. Using stochastic differential rules, the Hamiltonian process... satisfy: Related The stochastic differential equation can be written as: The drift coefficient and diffusion coefficient are as follows: ; ; The domain of the integral is as follows: The following transformations are introduced for the drift coefficient and diffusion coefficient: ; The expressions for the drift coefficient and diffusion coefficient can be transformed into: ; ; ; The domain of the integral is defined as follows: .

[0030] It should be noted that, as can be seen from the formula, the drift term coefficient and diffusion term coefficient An exact expression cannot be derived. Therefore, numerical methods can be used to calculate the drift diffusion coefficient. , .

[0031] Represents total energy. Represents the total energy of the system The limit value. Assume... In the interval Internal changes, when The system fails. Assume... The conditional reliability function is defined as follows: In time interval The interior is located in a certain safe area The probability, given an initial Hamiltonian. In a certain safe area Inside: Then, it can be proven that the conditional reliability function satisfies the following BK equation: Initial conditions exist: Boundary conditions: ; ;if and The boundary conditions become: ;if and The boundary conditions become: The probability distribution function for the first time travel is: The first time travel CPDF is: ;Initial and boundary conditions of the BK equation and The same as in the Hamiltonian form of the equations of motion, except Initial state Replaced by. Conditional probability density of the first time travel. The following can be obtained from the conditional reliability function: ;in, This is the first time travel, that is, in Under the conditions, First time reaching the critical value Random time.

[0032] It can be proven that, in order to further characterize the distribution characteristics of the first time travel, we derive statistical moments. Satisfy the following PG equation: MFPT is: MFPT Satisfying the PG equation: The boundary conditions for the PG equation are: ; ;in MFPT is subject to boundary conditions: ; The system's conditional reliability function is governed by the BK equations, and its specific boundary conditions are determined by... Definition. The BK equations are then solved using an implicit finite difference method based on the Crank-Nicolson scheme.

[0033] This embodiment takes a two-degree-of-freedom rotational vibration energy harvester as the research object and elaborates on the specific implementation process of the present invention in detail: setting the critical energy as... This condition indicates that when the system energy exceeds... The energy harvester will be damaged.

[0034] System parameters are set to , , , , , and .

[0035] In the analysis below, only the parameters discussed in the figure have changed; the other parameters remain constant. The drift coefficient and diffusion coefficient are crucial to the reliability of the two-degree-of-freedom rotating vibration energy harvesting system, and the drift coefficient and diffusion coefficient of this system are solvable.

[0036] Figure 2 (a)-(f) respectively gives noodle, noodle, noodle, , and The drift coefficient and diffusion coefficient of the surface are given in Figure 3 (a)-(f). noodle, noodle, noodle, , and Cross-sectional diagrams of the drift coefficient and diffusion coefficient of the surface. It can be seen that the diffusion is mainly influenced by the diffusion process. With initial energy Dominant, drift is primarily damped ( , ), generator frequency ( , ) and friction leading; The mode of action and Similar but through different channels. The overall behavior is "diffusion depends on energy and..." "Growth and drift become more negative with increasing energy and dissipation parameters," thus forming a convection-dominated one-dimensional energy diffusion. The drift and diffusion coefficients are represented on a three-dimensional surface in the six-parameter-energy plane. Orange represents the drift coefficient, and blue represents the diffusion coefficient. It can be seen that the diffusion coefficient monotonically increases with energy and is affected by noise intensity. It exhibits an approximately linear proportionality, while being approximately insensitive to damping and friction parameters; the drift coefficient decreases monotonically with energy, and is also affected by... , , and Significantly enhanced, for This only manifests as a slight overall upward shift. This result is in perfect agreement with the coefficient expression derived from the quasi-non-integrable Hamiltonian stochastic averaging, indicating that the system is convection-dominated in most energy ranges.

[0037] The effectiveness of the theoretical method is verified by comparing Monte Carlo (MC) simulations with the Stochastic Average Method (SAM): Figure 4 shows that the conditional reliability function solved by SAM and the MC simulation results are in high agreement in the time domain, proving the accuracy of the system reliability analysis based on the stochastic average method; Figure 5 shows that the CPDF of the first crossing time obtained by SAM and MC matches well in distribution trend, peak value, and decay law, verifying the theoretical reliability of the probabilistic characteristic analysis of the first crossing behavior; Figure 6 shows different initial energies... The results show a high degree of agreement between the MFPT calculated by SAM and the MC simulation results, further demonstrating the effectiveness of the stochastic averaging method in quantifying the time-scale characteristics of the system's first crossing failure, and providing methodological support for subsequent parameter analysis.

[0038] As shown in Figure 7, the initial energy intensity It has a significant modulating effect on the system's first-crossing reliability. As shown in Figure 7(a), the conditional reliability function changes with the initial energy intensity. The probability density function decreases as the initial energy increases. This phenomenon indicates that as the initial energy approaches the absorption boundary, the system is more likely to experience a first crossover event within a shorter time, leading to a decrease in overall reliability. Figure 7(b) shows the conditional probability density function of the first crossover, which decreases as the initial energy intensity increases. Increased initial energy levels, with a significantly higher peak density and a shift to a shorter timescale, indicate a significantly increased probability of initial energy overflow in the short term. In engineering applications, excessively high initial energy can be suppressed by optimizing startup procedures and energy management, or by configuring more conservative safety margins and maintenance strategies under high initial energy intensity conditions to reduce the risk of early failure.

[0039] As shown in Figure 8, the noise intensity D has a significant moderating effect on the first-crossing reliability of the system, where (a) and (b) in Figure 8 both select The value is 0.7. The conditional reliability function shown in Figure 8(a) decreases with increasing noise intensity D. The first pass CPDF in Figure 8(b) exhibits a unimodal distribution; the peak value of CPDF increases with increasing noise intensity D, indicating a significant increase in the short-time first pass probability. The MFPT in Figure 8(c) decreases with increasing noise intensity D. The results for the conditional reliability function, first pass time CPDF, and MFPT are highly consistent, indicating that increased noise intensity significantly reduces system time reliability, shortens service life, and exacerbates the risk of early failure. In engineering applications, early first pass events under high noise conditions can be suppressed through damping configuration, operational strategy optimization, or conservative monitoring thresholding.

[0040] Main system natural frequency The natural frequency is a key structural parameter affecting the reliability of the system's first crossing under random conditions. The influence of [the two factors] is relatively weak within the parameter range of this study; therefore, focusing on analyzing their differential effects is of engineering significance. Figures 9(a)(b) and 10(a)(b) both select [the appropriate parameters]. It is 0.7. As shown in Figure 9(a), with the natural frequency of the main system... As the frequency increases, the conditional reliability function decreases. In Figure 9(b), the CPDF of the first crossover time exhibits a unimodal distribution, decreasing with increasing master system natural frequency. As the value increases, the peak value of CPDF rises, enhancing the probability of early failure. The MFPT in Figure 9(c) increases with... Increase monotonically decreasing. Different values ​​in (a)-(c) of Figure 10. The corresponding conditional reliability function, the CPDF curve for the first crossing, and the MFPT curve do not change significantly. However, with the natural frequency... As the value increases, the conditional reliability function decreases, the peak value of the first crossover CPDF increases, and the MFPT decreases. The results for all three indices are consistent, indicating that the natural frequency of the main system... Increasing the frequency significantly reduces system time reliability, shortens service life, and exacerbates the risk of early failure. (The last sentence appears to be incomplete and possibly refers to a different topic.) Increasing the frequency also reduces system reliability, although the effect is relatively insignificant. In engineering applications, reliability assessments and robust design should prioritize targeting the natural frequency. Parameter tuning and tolerance control are performed, and the natural frequency is listed as a primary design variable. Treated as a secondary factor to improve the system's probabilistic service reliability and lifespan.

[0041] Generator natural frequency These are key design variables that affect the reliability of the system's first crossing under random environments; therefore, analyzing their role is of significant engineering importance. Figure 11(a) and (b) show the selected... The value is 0.7. As shown in Figure 11(a), the conditional reliability function varies with the generator's natural frequency. The value increases with the increase of the generator's natural frequency. In Figure 11(b), the first crossing of the CPDF shows a single-peak distribution, increasing with the generator's natural frequency. As the frequency increases, the peak value of CPDF decreases. Figure 11(c) shows the MFPT as a function of the generator's natural frequency. The reliability increases with increasing generator natural frequency. All three sets of indicators consistently show that system reliability increases with increasing generator natural frequency. In engineering applications, increasing the generator natural frequency can be achieved by optimizing rotor structure design or selecting high-rigidity materials.

[0042] Mechanical damping ratio It is a key parameter affecting the reliability of the system's first crossing under random conditions. It has a monotonically increasing stability effect on the system's reliability, so analyzing its role is of great engineering significance. Among them, (a) and (b) in Figure 12 are selected The reliability function is 0.7. Figure 12(a) shows the reliability function as a function of mechanical damping ratio. As the mechanical damping ratio increases, the reliability function shifts upwards overall, and the attenuation slope decreases. In Figure 12(b), the PDF of the first crossing time all exhibit a unimodal left-skewed distribution, increasing with the mechanical damping ratio. As the damping ratio increases, the peak value of PDF decreases, and the probability of short-term first penetration decreases. Figure 12(c) shows the MFPT as a function of mechanical damping ratio. The mechanical damping ratio increases with increasing damping. Conditional reliability functions, first-pass time PDF, and MFPT all yield the same conclusion: increasing the mechanical damping ratio significantly improves system time reliability and extends mean service life. In engineering applications, increasing the mechanical damping ratio can be achieved by selecting high-damping alloys, adding dampers, or applying viscoelastic damping layers to critical structural components.

[0043] Electromagnetic damping ratio Electromagnetic damping ratio plays a crucial role in characterizing the performance of energy harvesters under random environmental conditions; therefore, analyzing the electromagnetic damping ratio is essential. The impact on system reliability is necessary. Figure 13(a) and (b) show the selection... The value is 0.7. From the reliability function shown in Figure 13(a), we can derive that the conditional reliability function changes with the electromagnetic damping ratio. The conclusion is that the electromagnetic damping ratio increases with the increase of [the specific value]. As shown in Figure 13(b), [the electromagnetic damping ratio increases with the increase of [the specific value]. As the electromagnetic damping ratio increases, the peak value of CPDF decreases. This means that the probability of first-pass passage within a short period of time is low. Figure 13(c) shows that as the electromagnetic damping ratio increases, the peak value of CPDF decreases. As the electromagnetic damping ratio increases, the MFPT also increases. Conditional reliability function, first-pass time (CPDF), and MFPT all yield the same conclusion: appropriately increasing the electromagnetic damping ratio can improve the reliability of the energy harvester. In engineering applications, increasing the electromagnetic damping ratio can be achieved by adjusting the external load resistance of the generator, increasing the number of coil turns, or optimizing the coil winding method.

[0044] Coulomb friction Coulomb friction has a significant impact on the performance of energy harvesters. The effects on the conditional reliability function, first crossover time PDF, and MFPT are shown in Figure 14. Figure 14(a) and (b) are selected... It is 0.7. As can be seen from Figure 14(a), with Coulomb friction... As the coefficient of friction increases, the conditional reliability function increases. The results in Figure 14(b) show that as the coefficient of friction increases... As the coefficient of friction increases, the PDF peak decreases. This condition implies a lower probability of the first crossing within a short period. Figure 14(c) shows that as Coulomb friction increases... As the coefficient of friction (CFT) increases, the MFPT also increases. Conditional reliability functions, first-crossing time (PDF), and MFPT all lead to the same conclusion: increasing the Coulomb friction of the system can improve the reliability of the rotating vibration energy harvester. In engineering applications, increasing Coulomb friction can be achieved by selecting contact materials with a high coefficient of friction, adjusting the preload pressure at the contact interface, or optimizing the surface roughness of the contact surfaces.

[0045] As shown in Table 1, this paper conducts parameter sensitivity analysis by applying a uniform 20% perturbation to each parameter and quantifying them using the average elasticity index. Let a certain parameter be... Its benchmark value is The disturbance value is The corresponding reliability functions are denoted as follows: and At a given moment At this point, instantaneous elasticity is defined as: ;in, In this study there are To obtain a single dimensionless measure over the entire observation period, the absolute values ​​of the instantaneous elasticity at several representative moments are averaged to obtain the average elasticity index: ;in, The corresponding time is taken This normalization index characterizes the parameters. The "unit relative disturbance" is the average amplitude of the relative change in conditional reliability caused within a selected time range. Therefore, it allows for horizontal comparisons between parameters with different dimensions and physical meanings to determine their sensitivity levels. The elasticity index calculated based on the above definition shows that the influence of system conditional reliability on each parameter has a clear hierarchical structure.

[0046] Initial energy and noise intensity These two parameters are the most dominant, with their average elasticity being much higher than the others. This indicates that even small changes to these two parameters can cause significant changes in the reliability of the conditions: among them... Directly control the relative proximity between the initial energy state and the failure threshold, while It primarily regulates the fluctuation amplitude and diffusion rate during energy evolution under random excitation. The natural frequency of the linear principal system. Electromagnetic damping ratio and the generator's natural frequency The set of parameters constituting moderate sensitivity, with average elasticity roughly in the range of (0.3-0.6 s), primarily depends on the dynamic balance between stochastic excitation resonant coupling and energy dissipation efficiency in modulating system reliability. In contrast, the natural frequencies of the nonlinear master system... Mechanical damping ratio and Coulomb friction The sensitivity can be ignored This indicates that within the range of parameters under investigation, the system reliability is essentially insensitive to changes in these parameters. This hierarchical sensitivity structure provides a quantitative basis for parameter prioritization in reliability-oriented design, helping to focus on a limited number of key parameters in the optimized design of vibration kinetic energy harvesters, while reducing design redundancy associated with low-sensitivity parameters. It should be noted that the above sensitivity ranking comes from a local first-order sensitivity analysis conducted under specific operating conditions. Therefore, this ranking mainly characterizes the relative influence of each parameter within the neighborhood of that operating point, and should not be interpreted as a universally applicable conclusion for all operating conditions. Furthermore, the qualitative labels such as "high," "medium," and "low" sensitivity used in this paper are only for engineering purposes. The size is explained in a grading manner for easy understanding, rather than as a strict quantitative threshold.

[0047] Table 1

[0048] Experimental results show that appropriately increasing the generator's natural frequency, mechanical damping ratio, electromagnetic damping ratio, and Coulomb friction can improve system reliability, while increasing the initial energy intensity, noise intensity, and main system natural frequency will decrease system reliability. Parameter sensitivity analysis reveals a clear hierarchy of parameters' impact on system reliability, with initial energy and noise intensity being the most dominant parameters. The linear natural frequency of the main system, electromagnetic damping ratio, and generator frequency have a weaker impact on system reliability than initial energy and noise intensity, while the nonlinear natural frequency of the main system, mechanical damping ratio, and Coulomb friction have negligible impact on system reliability.

[0049] Therefore, this invention adopts the above-mentioned reliability assessment system and method for a two-degree-of-freedom rotating vibration energy harvester, combining nonlinear dynamic modeling, dimensionality reduction of high-dimensional systems, reliability index analysis, and parameter sensitivity analysis. It simplifies high-dimensional systems by using the stochastic averaging method for quasi-non-integrable Hamiltonian systems, avoiding complex calculations. Based on the BK and GP equations, it obtains indicators such as conditional reliability function, first crossover time PDF, and MFPT. Verification shows that the calculation results of this method are in good agreement with the simulation results, fully demonstrating its advantages in accuracy and efficiency in stochastic reliability analysis of high-dimensional nonlinear systems.

[0050] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester, characterized in that, Includes the following steps: S1. Dynamic Modeling: For a two-degree-of-freedom rotating vibration energy harvester containing a main oscillator and a rotor branch, a nonlinear dynamic model considering Coulomb friction terms is established, the system parameters are defined, and the equations of motion of the dynamic model are transformed into Hamiltonian form through variable transformation; S2. Stochastic Averaging Dimensionality Reduction: The quasi-nonintegrable Hamiltonian system obtained in step S1 is subjected to stochastic averaging method, and the rapidly changing variables are averaged over time to simplify the high-dimensional coupled system into a one-dimensional stochastic energy diffusion process containing only total energy, so as to preserve the key characteristics of the system's energy evolution; S3. Calculation of Drift and Diffusion Coefficients: Based on the one-dimensional energy diffusion process described in step S2, the drift coefficient, diffusion coefficient, and energy period of the energy diffusion process are calculated through periodic integration and numerical integration in the energy domain. When the closed-form analytical solution is unavailable, high-precision numerical integration is used for estimation. S4. Solving and Verification of Reliability Indicators: Using the energy threshold as the failure criterion, backward Kolmogorov equations and generalized Pontryagin equations are constructed in the energy domain. The boundary value problem is solved in combination with the corresponding initial and boundary conditions to obtain the conditional reliability function, the conditional probability density function CPDF of the first crossing time, and the mean first crossing time MFPT. The calculation results of the conditional reliability function, the first crossing time CPDF, and MFPT are compared and verified by Monte Carlo simulation. S5. Parameter Sensitivity Analysis and Design Optimization: Based on the reliability index obtained in step S4, a parameter set including initial energy, noise intensity, natural frequency, mechanical damping ratio, electromagnetic damping ratio, and Coulomb friction coefficient is selected. A perturbation of a predetermined amplitude is applied to each parameter in the parameter set, and the instantaneous elastic index is defined and calculated. The average elastic index at multiple representative moments is obtained, thereby quantifying the sensitivity level of each parameter and forming an engineering adjustment and optimization scheme for reliability design.

2. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 1, characterized in that, The dynamic model includes the main oscillator and the rotor branch, and the equation of motion is expressed as follows: ； ;in, For quality, The mechanical damping coefficient is... The electromagnetic damping coefficient is... 、 All are linear stiffness coefficients. The stiffness coefficient is a third-order nonlinear coefficient. For quality The relative displacement between the base and the support. For generator The relative displacement between the base and the support. It is the inertial constant. Let be the value of the Coulomb friction force. It is the acceleration of the base; the Hamiltonian equation of motion is: ； ;in, , , , ， and For the damping ratio, 、 、 It is the system frequency. is the Coulomb friction coefficient.

3. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 1, characterized in that, The stochastic averaging method performs time averaging on rapidly changing variables to eliminate high-frequency oscillations: The total energy is defined as follows: ;in, The energy of the main oscillator The energy is in the rotor branch; the expression for the one-dimensional energy diffusion process is: ;in, The drift coefficient, The diffusion coefficient is... This is standard Brownian motion.

4. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 1, characterized in that, The expressions for the drift coefficient and diffusion coefficient are as follows: ； ;in, For energy cycles, The domain of integration is defined as follows: The energy period expression is: The expression for the domain of the integral is: 。 5. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 1, characterized in that, The drift coefficient, diffusion coefficient, and energy period of the energy diffusion process are given by the energy domain integral, and high-precision numerical integral estimation is used when the closed-form solution is unavailable.

6. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 1, characterized in that, The failure criterion sets the upper boundary of the energy domain as an absorbing boundary and the lower boundary as a reflecting or natural boundary; the equation for the backward Kolmogorov (BK) is: The equation for the generalized Pontryagin (PG) is: The conditional reliability function expression is: The CPDF expression for the first time travel is: The MFPT expression is: ;MFPT satisfies the PG equation: ;MFPT is given by mutual proof of the BK equation and the PG equation.

7. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 1, characterized in that, The parameter sensitivity analysis includes: selecting a set of sensitive parameters covering initial energy, noise intensity, natural frequency, nonlinear stiffness coefficient, mechanical damping ratio, electromagnetic damping ratio, and Coulomb friction intensity; quantification method: applying a 20% perturbation to each parameter in the parameter set, defining an instantaneous elastic index, taking the average elastic index at multiple times, and ranking the parameter sensitivity levels.

8. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 7, characterized in that, The instantaneous elasticity index is expressed as follows: ;in, For parameters, The reference value for the parameter is... The reliability function for the parameters. This is a reliability function for the parameter baseline value.

9. The reliability assessment method for a two-degree-of-freedom rotational vibration energy harvester according to claim 1, characterized in that, The engineering parameter adjustment recommendations include: providing an initial energy safety margin control strategy for the installation and startup phases; and providing a coordinated design strategy for vibration isolation / damping and lubrication for random strong excitation scenarios.

10. An evaluation system for implementing the reliability evaluation method for a two-degree-of-freedom rotating vibration energy harvester as described in claims 1-9, characterized in that, include: The system includes a processor and a memory, the memory storing a computer program executable on the processor. When executing the computer program, the processor is configured as a dynamic modeling module to establish a nonlinear dynamic model of a two-degree-of-freedom rotating vibration energy harvester, incorporating Coulomb friction terms and transforming the equations of motion of the dynamic model into Hamiltonian form. A stochastic averaging dimensionality reduction module is used to apply a stochastic averaging method to the quasi-nonintegrable system in Hamiltonian form, reducing the high-dimensional coupled system to a one-dimensional energy diffusion process containing only total energy, and evaluating the drift coefficient, diffusion coefficient, and energy period of the energy diffusion process. A first-crossing index is also included. The calculation module is used to construct and solve the boundary value problems of the backward Kolmogorov equation and the generalized Pontryagin equation in the energy domain, obtaining the conditional reliability function, the CPDF of the first crossing time, and the MFPT. The calculation results can be compared and verified with Monte Carlo simulation results. The parameter sensitivity analysis and optimization module is used to calculate the average elastic sensitivity index based on the conditional reliability function and the first crossing time statistics, rank the sensitivity of initial energy, noise intensity, natural frequency, mechanical damping ratio, electromagnetic damping ratio, and Coulomb friction parameters, and output reliability-oriented engineering adjustment and design optimization schemes.

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