研究目的
Investigating the modes of vibration, stability, and internal resonances of non-linear piezoelectric small-scale beams using modified couple stress theory.
研究成果
The study successfully investigates non-linear free vibrations of piezoelectric small-scale beams using modified couple stress theory. Key findings include the identification of hardening spring behavior, increased stiffness due to size effects, and the discovery of secondary branches from bifurcations due to internal resonances (e.g., 1:4 and 1:10 resonances). The shooting method proves effective for stability analysis and capturing complex dynamics, while the multiple scales method is valid only for smaller amplitudes. The piezoelectric effect has a minor impact on vibration characteristics. The work provides novel insights into the non-linear modes and stability of micro/nano beams, with implications for MEMS/NEMS applications.
研究不足
The model assumes specific boundary conditions (hinged-hinged) and neglects damping, which may affect stability analysis. The shooting method requires significant computational effort for multi-degree-of-freedom systems. The multiple scales method is limited to small vibration amplitudes and may not be accurate for large amplitudes. The study focuses on free vibrations without external excitation, limiting applicability to forced vibration scenarios. The piezoelectric effect is found to have a small influence, which might not hold for all materials or configurations.
1：Experimental Design and Method Selection:
The study employs a non-linear piezoelectric size-dependent Rayleigh’s beam model based on modified couple stress theory. Hamilton's principle is used to derive partial differential governing equations and boundary conditions. The method of multiple scales (an analytical perturbation method) and a shooting method (a numerical method for solving boundary value problems) are applied to solve the equations of motion. Stability analysis is conducted using Floquet multipliers from the monodromy matrix.
2：Sample Selection and Data Sources:
A hinged-hinged piezoelectric beam is chosen as an example. Material properties are based on published data, with specific values for piezoelectric coefficients, elastic moduli, and density provided in the paper.
3：List of Experimental Equipment and Materials:
No specific experimental equipment is mentioned as the study is theoretical and computational; it involves mathematical modeling and numerical simulations.
4：Experimental Procedures and Operational Workflow:
The governing equations are discretized using Galerkin's method to obtain ordinary differential equations. The shooting method is used to find periodic solutions by treating the equations as a two-point boundary value problem with periodicity conditions. Runge-Kutta-Fehlberg method with Cash-Karp adaptive step size is employed for integration. The method of multiple scales is applied for analytical approximations.
5：Data Analysis Methods:
Solutions are analyzed for stability using eigenvalues of the monodromy matrix. Backbone curves (relating frequency to amplitude) are plotted. Fourier spectra, phase-plane diagrams, and vibration shapes are examined to study bifurcations and internal resonances.
独家科研数据包，助您复现前沿成果，加速创新突破
获取完整内容
