The Carter model is used to characterize the dynamic behaviors of fracture growth and fracturing fluid leakoff.A thermo-fluid coupling temperature response forward model is built considering the fluid flow and heat tr...The Carter model is used to characterize the dynamic behaviors of fracture growth and fracturing fluid leakoff.A thermo-fluid coupling temperature response forward model is built considering the fluid flow and heat transfer in wellbore,fracture and reservoir.The influences of fracturing parameters and fracture parameters on the responses of distributed temperature sensing(DTS)are analyzed,and a diagnosis method of fracture parameters is presented based on the simulated annealing algorithm.A field case study is introduced to verify the model’s reliability.Typical V-shaped characteristics can be observed from the DTS responses in the multi-cluster fracturing process,with locations corresponding to the hydraulic fractures.The V-shape depth is shallower for a higher injection rate and longer fracturing and shut-in time.Also,the V-shape is wider for a higher fracture-surface leakoff coefficient,longer fracturing time and smaller fracture width.Additionally,the cooling effect near the wellbore continues to spread into the reservoir during the shut-in period,causing the DTS temperature to decrease instead of rise.Real-time monitoring and interpretation of DTS temperature data can help understand the fracture propagation during fracturing operation,so that immediate measures can be taken to improve the fracturing performance.展开更多
The Bayesian method of statistical analysis has been applied to the parameter identification problem. A method is presented to identify parameters of dynamic models with the Bayes estimators of measurement frequencies...The Bayesian method of statistical analysis has been applied to the parameter identification problem. A method is presented to identify parameters of dynamic models with the Bayes estimators of measurement frequencies. This is based on the solution of an inverse generalized evaluate problem. The stochastic nature of test data is considered and a normal distribution is used for the measurement frequencies. An additional feature is that the engineer's confidence in the measurement frequencies is quantified and incorporated into the identification procedure. A numerical example demonstrates the efficiency of the method.展开更多
The dynamic performance of a nozzle-flapper servo valve can be affected by several factors such as the disturbance of the input signal,the motion of the armature assembly and the oscillation of the jet force.As the pa...The dynamic performance of a nozzle-flapper servo valve can be affected by several factors such as the disturbance of the input signal,the motion of the armature assembly and the oscillation of the jet force.As the part of vibrating at high frequency,the armature assembly plays a vital role during the operation of the servo valve.In order to accurately predict the transient response of the armature assembly during the vibration,a mathematical model of armature assembly is established based on the distributed parameters method(DPM)and Hamilton principle.The new mathematical model is composed of three main parts,the modal eigenfunction,modal mechanical response expressions of the spring tube and the motion equation of the other armature assembly.After programing,the purpose of using the DPM to predict the dynamic response of different positions located on the armature assembly is achieved.For verifying the validity of the mathematical model,the finite element method(FEM)and classic model(CM)of armature assembly are applicated by commercial software under the same condition.The comparison results prove that the DPM can effectively predict the axial and tangential deflection of the armature assembly different positions which the CM can’t duing to its over-simplification.A certain error is generated when predicting the axial deformation at different heights by DPM,which is caused by an approximate method to simulate the torsion of the spring tube.The comparison results of the spring tube deflection at different vibration frequencies shows that the adaptability of DPM is significantly higher than the classic model,which verify the model is more adaptable for predicting the dynamic response of the armature assembly.展开更多
The complex geometrical features of mechanical components significantly influence contact interactions and system dynamics.However,directly modeling contact forces on surfaces with intricate geometries presents consid...The complex geometrical features of mechanical components significantly influence contact interactions and system dynamics.However,directly modeling contact forces on surfaces with intricate geometries presents considerable challenges.This study focuses on the helically twisted wire rope-sheave contact and proposes a contact force model that incorporates complex geometric features through a parameter identification approach.The model's impact on contact forces and system dynamics is thoroughly investigated.Leveraging a point contact model and an elliptic integral approximation,a loss function is formulated using the finite element(FE)contact model results as the reference data.Geometric parameters are subsequently determined by optimizing this loss function via a genetic algorithm(GA).The findings reveal that the contact stiffness increases with the wire rope pitch length,the radius of principal curvature,and the elliptic eccentricity of the contact zone.The proposed contact force model is integrated into a rigid-flexible coupled dynamics model,developed by the absolute node coordinate formulation,to examine the effects of contact geometry on system dynamics.The results demonstrate that the variations in wire rope geometry alter the contact stiffness,which in turn affects dynamic rope tension through frictional energy dissipation.The enhanced model's predictions exhibit superior alignment with the experimental data,thereby validating the methodology.This approach provides new insights for deducing the contact geometry from kinetic parameters and monitoring the performance degradation of mechanical components.展开更多
A novel distributed model predictive control scheme based on dynamic integrated system optimization and parameter estimation （DISOPE） was proposed for nonlinear cascade systems under network environment. Under the d...A novel distributed model predictive control scheme based on dynamic integrated system optimization and parameter estimation （DISOPE） was proposed for nonlinear cascade systems under network environment. Under the distributed control structure, online optimization of the cascade system was composed of several cascaded agents that can cooperate and exchange information via network communication. By iterating on modified distributed linear optimal control problems on the basis of estimating parameters at every iteration the correct optimal control action of the nonlinear model predictive control problem of the cascade system could be obtained, assuming that the algorithm was convergent. This approach avoids solving the complex nonlinear optimization problem and significantly reduces the computational burden. The simulation results of the fossil fuel power unit are illustrated to verify the effectiveness and practicability of the proposed algorithm.展开更多
Industrial noise can be successfully mitigated with the combined use of passive and Active Noise Control (ANC) strategies. In a noisy area, a practical solution for noise attenuation may include both the use of baffle...Industrial noise can be successfully mitigated with the combined use of passive and Active Noise Control (ANC) strategies. In a noisy area, a practical solution for noise attenuation may include both the use of baffles and ANC. When the operator is required to stay in movement in a delimited spatial area, conventional ANC is usually not able to adequately cancel the noise over the whole area. New control strategies need to be devised to achieve acceptable spatial coverage. A three-dimensional actuator model is proposed in this paper. Active Noise Control (ANC) usually requires a feedback noise measurement for the proper response of the loop controller. In some situations, especially where the real-time tridimensional positioning of a feedback transducer is unfeasible, the availability of a 3D precise noise level estimator is indispensable. In our previous works [1,2], using a vibrating signal of the primary source of noise as an input reference for spatial noise level prediction proved to be a very good choice. Another interesting aspect observed in those previous works was the need for a variable-structure linear model, which is equivalent to a sort of a nonlinear model, with unknown analytical equivalence until now. To overcome this in this paper we propose a model structure based on an Artificial Neural Network (ANN) as a nonlinear black-box model to capture the dynamic nonlinear behaveior of the investigated process. This can be used in a future closed loop noise cancelling strategy. We devise an ANN architecture and a corresponding training methodology to cope with the problem, and a MISO (Multi-Input Single-Output) model structure is used in the identification of the system dynamics. A metric is established to compare the obtained results with other works elsewhere. The results show that the obtained model is consistent and it adequately describes the main dynamics of the studied phenomenon, showing that the MISO approach using an ANN is appropriate for the simulation of the investigated process. A clear conclusion is reached highlighting the promising results obtained using this kind of modeling for ANC.展开更多
To achieve the manufacturing of Thin-Wall and High-Rib Components(TWHRC)with high precision,a novel heavy load Multi-DOF Envelope Forming Press(MEFP)with Parallel Kinematic Mechanism(PKM),driven by six Permanent Magne...To achieve the manufacturing of Thin-Wall and High-Rib Components(TWHRC)with high precision,a novel heavy load Multi-DOF Envelope Forming Press(MEFP)with Parallel Kinematic Mechanism(PKM),driven by six Permanent Magnet Synchronous Motors(PMSMs),is developed.However,on account of the heavy forming load,the PMSM parameters are in great variation.Meanwhile,the PMSM is always in a transient state caused by fast time-varying forming load,resulting in low identification precision of varied PMSM parameters and control precision of PMSM under traditional parameter identification methods.To solve this problem,a novel Sliding Mode Control Method with Enhanced PMSM Parameter Identification(SMCMEPPI)for heavy load MEFP is proposed.Firstly,the kinematic model of MEFP is established.Secondly,the variation law of PMSM parameters under heavy load is revealed.Thirdly,an enhanced PMSM parameter identification method is proposed,in which the q axis current of PMSM is used to represent the changing rate of forming load and the adjustment factor is first proposed to remove improper input of PMSM parameter identification online.Fourthly,the Electromechanical Coupling Dynamic Model(ECDM)of MEFP,which includes identified PMSM parameters,is developed.Finally,based on the developed ECDM,a novel SMCMEPPI is proposed to realize the high-precision control of heavy load MEFP.The experimental results indicate that the proposed SMCMEPPI can significantly improve the control precision of heavy load MEFP.展开更多
为准确描述下降管反应器内生物质颗粒与高温陶瓷球之间的瞬态传热与热解行为,该研究提出了一种基于分布活化能模型(distributed activation energy model,DAEM)的多物理场耦合数值模型。该模型在颗粒能量平衡框架下,引入接触导热、气膜...为准确描述下降管反应器内生物质颗粒与高温陶瓷球之间的瞬态传热与热解行为,该研究提出了一种基于分布活化能模型(distributed activation energy model,DAEM)的多物理场耦合数值模型。该模型在颗粒能量平衡框架下,引入接触导热、气膜导热、对流与辐射多种传热机制,并与质量转化过程和活化能分布特征相耦合,建立了用于描述生物质快速热解过程的常微分方程模型。基于热重分析试验数据,对高斯、洛伦兹及逻辑斯谛3种活化能分布函数进行了参数反演与对比分析。结果表明,洛伦兹分布能够更准确地再现试验热重曲线,其平均绝对误差(mean absolute error,MAE)和均方根误差(root mean square error,RMSE)分别为0.0116和0.0138。数值模拟结果显示,生物质颗粒在初始阶段经历了极高的升温速率(峰值达到2.14×10^(3)℃/s),但热解反应相对于温度演化存在明显的动力学滞后特征。传热机制分析表明,对流与导热在整个热解过程中占主导地位,而在高温阶段辐射传热的贡献不可忽略。参数敏感性分析进一步揭示,陶瓷球温度和生物质颗粒粒径对热解效率具有显著影响,反应焓和颗粒碰撞概率次之,而辐射视角因子的影响相对有限。研究结果表明,在传热条件充分的快速热解工况下,过程控制机理由传热受限逐渐转变为化学反应动力学受限。研究为深入理解下降管反应器内多物理场耦合热解行为特征及反应器结构与工艺参数优化提供了理论依据和数据支持。展开更多
Permanent magnet synchronous motors(PMSMs)are typical electromechanical energy-conversion systems,in which the electrical and mechanical subsystems interact and impact each other.However,existing studies have investig...Permanent magnet synchronous motors(PMSMs)are typical electromechanical energy-conversion systems,in which the electrical and mechanical subsystems interact and impact each other.However,existing studies have investigated these two subsystems independently and failed to determine the coupling effect between electrical signals and mechanical vibrations.To address these gaps,a comprehensive electromechanical coupled model is proposed herein.This model integrates the PMSM model based on the winding function and the rotor-bearing dynamics model.The developed model can take into account the variations in inductance and current caused by non-uniform air-gap distribution.The electromechanical dynamic responses of the PMSM under rotor-bearing vibration and rotor eccentricity conditions are systematically analysed using this model.Results demonstrate that the proposed model improved the accuracy of both internal and external excitation representation in PMSMs compared with the conventional models.The dynamic behaviour of the rotor-bearing system is distinctly reflected in the electrical signals,and the variation laws of rotor eccentric distance and eccentric angle on the dynamic characteristics of the PMSM are revealed.The proposed model provides theoretical support for investigating the electromechanical coupled effect in PMSMs and offers an effective approach for state detection and fault diagnosis of motor-driven systems.展开更多
The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Eul...The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Euler-Bernoulli model with inextensible deformation. A nonlinear distributed parameter model of cantilevered piezoelectric energy harvesters is proposed using the generalized Hamilton＇s principle. The proposed model includes geometric and inertia nonlinearity, but neglects the material nonlinearity. Using the Galerkin decomposition method and harmonic balance method, analytical expressions of the frequency-response curves are presented when the first bending mode of the beam plays a dominant role. Using these expressions, we investigate the effects of the damping, load resistance, electromechanical coupling, and excitation amplitude on the frequency-response curves. We also study the difference between the nonlinear lumped-parameter and distributed- parameter model for predicting the performance of the energy harvesting system. Only in the case of parametric excitation, we demonstrate that the energy harvesting system has an initiation excitation threshold below which no energy can be harvested. We also illustrate that the damping and load resistance affect the initiation excitation threshold.展开更多
基金Supported by the National High-Tech Research Project(GJSCB-HFGDY-2024-004)National Natural Science Foundation of China(12402305)+2 种基金Postdoctoral Fellowship Program of CPSF(GZC20232200)China Postdoctoral Science Foundation(2024M762703)Sichuan Science and Technology Program(2025ZNSFSC1352)。
文摘The Carter model is used to characterize the dynamic behaviors of fracture growth and fracturing fluid leakoff.A thermo-fluid coupling temperature response forward model is built considering the fluid flow and heat transfer in wellbore,fracture and reservoir.The influences of fracturing parameters and fracture parameters on the responses of distributed temperature sensing(DTS)are analyzed,and a diagnosis method of fracture parameters is presented based on the simulated annealing algorithm.A field case study is introduced to verify the model’s reliability.Typical V-shaped characteristics can be observed from the DTS responses in the multi-cluster fracturing process,with locations corresponding to the hydraulic fractures.The V-shape depth is shallower for a higher injection rate and longer fracturing and shut-in time.Also,the V-shape is wider for a higher fracture-surface leakoff coefficient,longer fracturing time and smaller fracture width.Additionally,the cooling effect near the wellbore continues to spread into the reservoir during the shut-in period,causing the DTS temperature to decrease instead of rise.Real-time monitoring and interpretation of DTS temperature data can help understand the fracture propagation during fracturing operation,so that immediate measures can be taken to improve the fracturing performance.
文摘The Bayesian method of statistical analysis has been applied to the parameter identification problem. A method is presented to identify parameters of dynamic models with the Bayes estimators of measurement frequencies. This is based on the solution of an inverse generalized evaluate problem. The stochastic nature of test data is considered and a normal distribution is used for the measurement frequencies. An additional feature is that the engineer's confidence in the measurement frequencies is quantified and incorporated into the identification procedure. A numerical example demonstrates the efficiency of the method.
基金supported by National Natural Science Foundation of China(No.51675119)。
文摘The dynamic performance of a nozzle-flapper servo valve can be affected by several factors such as the disturbance of the input signal,the motion of the armature assembly and the oscillation of the jet force.As the part of vibrating at high frequency,the armature assembly plays a vital role during the operation of the servo valve.In order to accurately predict the transient response of the armature assembly during the vibration,a mathematical model of armature assembly is established based on the distributed parameters method(DPM)and Hamilton principle.The new mathematical model is composed of three main parts,the modal eigenfunction,modal mechanical response expressions of the spring tube and the motion equation of the other armature assembly.After programing,the purpose of using the DPM to predict the dynamic response of different positions located on the armature assembly is achieved.For verifying the validity of the mathematical model,the finite element method(FEM)and classic model(CM)of armature assembly are applicated by commercial software under the same condition.The comparison results prove that the DPM can effectively predict the axial and tangential deflection of the armature assembly different positions which the CM can’t duing to its over-simplification.A certain error is generated when predicting the axial deformation at different heights by DPM,which is caused by an approximate method to simulate the torsion of the spring tube.The comparison results of the spring tube deflection at different vibration frequencies shows that the adaptability of DPM is significantly higher than the classic model,which verify the model is more adaptable for predicting the dynamic response of the armature assembly.
基金supported by the National Key Research and Development Program of China(No.2023YFC3010400)。
文摘The complex geometrical features of mechanical components significantly influence contact interactions and system dynamics.However,directly modeling contact forces on surfaces with intricate geometries presents considerable challenges.This study focuses on the helically twisted wire rope-sheave contact and proposes a contact force model that incorporates complex geometric features through a parameter identification approach.The model's impact on contact forces and system dynamics is thoroughly investigated.Leveraging a point contact model and an elliptic integral approximation,a loss function is formulated using the finite element(FE)contact model results as the reference data.Geometric parameters are subsequently determined by optimizing this loss function via a genetic algorithm(GA).The findings reveal that the contact stiffness increases with the wire rope pitch length,the radius of principal curvature,and the elliptic eccentricity of the contact zone.The proposed contact force model is integrated into a rigid-flexible coupled dynamics model,developed by the absolute node coordinate formulation,to examine the effects of contact geometry on system dynamics.The results demonstrate that the variations in wire rope geometry alter the contact stiffness,which in turn affects dynamic rope tension through frictional energy dissipation.The enhanced model's predictions exhibit superior alignment with the experimental data,thereby validating the methodology.This approach provides new insights for deducing the contact geometry from kinetic parameters and monitoring the performance degradation of mechanical components.
基金This work was supportedbytheNationalNaturalScienceFoundationofChina(No.60474051),theProgramforNewCenturyExcellentTalentsinUniversityofChina(NCET),andtheSpecializedResearchFundfortheDoctoralProgramofHigherEducationofChina(No.20020248028).
文摘A novel distributed model predictive control scheme based on dynamic integrated system optimization and parameter estimation （DISOPE） was proposed for nonlinear cascade systems under network environment. Under the distributed control structure, online optimization of the cascade system was composed of several cascaded agents that can cooperate and exchange information via network communication. By iterating on modified distributed linear optimal control problems on the basis of estimating parameters at every iteration the correct optimal control action of the nonlinear model predictive control problem of the cascade system could be obtained, assuming that the algorithm was convergent. This approach avoids solving the complex nonlinear optimization problem and significantly reduces the computational burden. The simulation results of the fossil fuel power unit are illustrated to verify the effectiveness and practicability of the proposed algorithm.
基金CAPES and CNPq(Brazilian federal research agencies)for their financial support.
文摘Industrial noise can be successfully mitigated with the combined use of passive and Active Noise Control (ANC) strategies. In a noisy area, a practical solution for noise attenuation may include both the use of baffles and ANC. When the operator is required to stay in movement in a delimited spatial area, conventional ANC is usually not able to adequately cancel the noise over the whole area. New control strategies need to be devised to achieve acceptable spatial coverage. A three-dimensional actuator model is proposed in this paper. Active Noise Control (ANC) usually requires a feedback noise measurement for the proper response of the loop controller. In some situations, especially where the real-time tridimensional positioning of a feedback transducer is unfeasible, the availability of a 3D precise noise level estimator is indispensable. In our previous works [1,2], using a vibrating signal of the primary source of noise as an input reference for spatial noise level prediction proved to be a very good choice. Another interesting aspect observed in those previous works was the need for a variable-structure linear model, which is equivalent to a sort of a nonlinear model, with unknown analytical equivalence until now. To overcome this in this paper we propose a model structure based on an Artificial Neural Network (ANN) as a nonlinear black-box model to capture the dynamic nonlinear behaveior of the investigated process. This can be used in a future closed loop noise cancelling strategy. We devise an ANN architecture and a corresponding training methodology to cope with the problem, and a MISO (Multi-Input Single-Output) model structure is used in the identification of the system dynamics. A metric is established to compare the obtained results with other works elsewhere. The results show that the obtained model is consistent and it adequately describes the main dynamics of the studied phenomenon, showing that the MISO approach using an ANN is appropriate for the simulation of the investigated process. A clear conclusion is reached highlighting the promising results obtained using this kind of modeling for ANC.
基金the National Science and Technology Major Project of China(No.2019-Ⅶ-0017-0158)the National Natural Science Foundation of China(Nos.U2037204,U21A20131)the Innovative Research Team Development Program of Ministry of Education of China(No.IRT17R83)for the support given to this research。
文摘To achieve the manufacturing of Thin-Wall and High-Rib Components(TWHRC)with high precision,a novel heavy load Multi-DOF Envelope Forming Press(MEFP)with Parallel Kinematic Mechanism(PKM),driven by six Permanent Magnet Synchronous Motors(PMSMs),is developed.However,on account of the heavy forming load,the PMSM parameters are in great variation.Meanwhile,the PMSM is always in a transient state caused by fast time-varying forming load,resulting in low identification precision of varied PMSM parameters and control precision of PMSM under traditional parameter identification methods.To solve this problem,a novel Sliding Mode Control Method with Enhanced PMSM Parameter Identification(SMCMEPPI)for heavy load MEFP is proposed.Firstly,the kinematic model of MEFP is established.Secondly,the variation law of PMSM parameters under heavy load is revealed.Thirdly,an enhanced PMSM parameter identification method is proposed,in which the q axis current of PMSM is used to represent the changing rate of forming load and the adjustment factor is first proposed to remove improper input of PMSM parameter identification online.Fourthly,the Electromechanical Coupling Dynamic Model(ECDM)of MEFP,which includes identified PMSM parameters,is developed.Finally,based on the developed ECDM,a novel SMCMEPPI is proposed to realize the high-precision control of heavy load MEFP.The experimental results indicate that the proposed SMCMEPPI can significantly improve the control precision of heavy load MEFP.
文摘为准确描述下降管反应器内生物质颗粒与高温陶瓷球之间的瞬态传热与热解行为,该研究提出了一种基于分布活化能模型(distributed activation energy model,DAEM)的多物理场耦合数值模型。该模型在颗粒能量平衡框架下,引入接触导热、气膜导热、对流与辐射多种传热机制,并与质量转化过程和活化能分布特征相耦合,建立了用于描述生物质快速热解过程的常微分方程模型。基于热重分析试验数据,对高斯、洛伦兹及逻辑斯谛3种活化能分布函数进行了参数反演与对比分析。结果表明,洛伦兹分布能够更准确地再现试验热重曲线,其平均绝对误差(mean absolute error,MAE)和均方根误差(root mean square error,RMSE)分别为0.0116和0.0138。数值模拟结果显示,生物质颗粒在初始阶段经历了极高的升温速率(峰值达到2.14×10^(3)℃/s),但热解反应相对于温度演化存在明显的动力学滞后特征。传热机制分析表明,对流与导热在整个热解过程中占主导地位,而在高温阶段辐射传热的贡献不可忽略。参数敏感性分析进一步揭示,陶瓷球温度和生物质颗粒粒径对热解效率具有显著影响,反应焓和颗粒碰撞概率次之,而辐射视角因子的影响相对有限。研究结果表明,在传热条件充分的快速热解工况下,过程控制机理由传热受限逐渐转变为化学反应动力学受限。研究为深入理解下降管反应器内多物理场耦合热解行为特征及反应器结构与工艺参数优化提供了理论依据和数据支持。
基金supported by the National Natural Science Foundation of China(Grant Nos.52275132,52388102)the National Key R&D Program of China(Grant No.2022YFB3402100)the Sichuan Science and Technology Program(Grant No.2024NSFTD0011)。
文摘Permanent magnet synchronous motors(PMSMs)are typical electromechanical energy-conversion systems,in which the electrical and mechanical subsystems interact and impact each other.However,existing studies have investigated these two subsystems independently and failed to determine the coupling effect between electrical signals and mechanical vibrations.To address these gaps,a comprehensive electromechanical coupled model is proposed herein.This model integrates the PMSM model based on the winding function and the rotor-bearing dynamics model.The developed model can take into account the variations in inductance and current caused by non-uniform air-gap distribution.The electromechanical dynamic responses of the PMSM under rotor-bearing vibration and rotor eccentricity conditions are systematically analysed using this model.Results demonstrate that the proposed model improved the accuracy of both internal and external excitation representation in PMSMs compared with the conventional models.The dynamic behaviour of the rotor-bearing system is distinctly reflected in the electrical signals,and the variation laws of rotor eccentric distance and eccentric angle on the dynamic characteristics of the PMSM are revealed.The proposed model provides theoretical support for investigating the electromechanical coupled effect in PMSMs and offers an effective approach for state detection and fault diagnosis of motor-driven systems.
基金supported by the National Natural Science Foundation of China (Grant 11172087)
文摘The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Euler-Bernoulli model with inextensible deformation. A nonlinear distributed parameter model of cantilevered piezoelectric energy harvesters is proposed using the generalized Hamilton＇s principle. The proposed model includes geometric and inertia nonlinearity, but neglects the material nonlinearity. Using the Galerkin decomposition method and harmonic balance method, analytical expressions of the frequency-response curves are presented when the first bending mode of the beam plays a dominant role. Using these expressions, we investigate the effects of the damping, load resistance, electromechanical coupling, and excitation amplitude on the frequency-response curves. We also study the difference between the nonlinear lumped-parameter and distributed- parameter model for predicting the performance of the energy harvesting system. Only in the case of parametric excitation, we demonstrate that the energy harvesting system has an initiation excitation threshold below which no energy can be harvested. We also illustrate that the damping and load resistance affect the initiation excitation threshold.
