Abstract In this paper, the stress-driven <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\left(\sigma D\right)\)</EquationSource> </InlineEquation>and strain-driven <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\left(\varepsilon D\right)\)</EquationSource> </InlineEquation>two-phase local/nonlocal integral models(TPNIM) are employed to formulate the size-dependent nonlinear free vibration of functionally graded (FG) Timoshenko microbeams in a thermal environment. The nonlinear differential governing equations and boundary conditions of motion are derived based on Hamilton’s principle andVon Karman's nonlinearity. The integral constitutive equations between strain and nonlocal stress components are transformed unitedly into equivalent differential forms with constitutive constraints. Neglecting the nonlinear items, the linear vibration frequencies and corresponding linear vibration mode shapes (LVMS) can be determined through the Laplace transformation technique. Based on LVMS, the nonlinear free vibration frequency is expressed explicitly as a function of vibration amplitude through the Ritz-Galerkin technique (RGT). Meanwhile, the numerical method based on the generalized differential quadrature method (GDQM) together with Newton’s iterative process is directly employed to calculate the nonlinear free vibration frequencies for different vibration amplitude. The effects of the relevant parameters on the linear and nonlinear vibration frequencies are investigated theoretically and numerically for microbeams. Purpose In this paper, the stress-driven and strain-driven two-phase local/nonlocal integral models(TPNIM) are employed to formulate the size-dependent nonlinear free vibration of functionally graded (FG) Timoshenko microbeams in a thermal environment. Methods The nonlinear differential governing equations and boundary conditions of motion are derived based on Hamilton’s principle and Von Karman's nonlinearity. The integral constitutive equations between strain and nonlocal stress components are transformed unitedly into equivalent differential forms with constitutive constraints. Neglecting the nonlinear items, the linear vibration frequencies and corresponding linear vibration mode shapes (LVMS) can be determined through the Laplace transformation technique. Based on LVMS, the nonlinear free vibration frequency is expressed explicitly as a function of vibration amplitude through the Ritz-Galerkin technique (RGT). Meanwhile, the numerical method based on the generalized differential quadrature method (GDQM) together with Newton’s iterative process is directly employed to calculate the nonlinear free vibration frequencies for different vibration amplitude. Results The effects of the FG-index, environmental temperature variation and nonlocal length-scale parameters on the linear and nonlinear vibration frequencies are investigated theoretically and numerically for microbeams under different boundary conditions. Conclusion The accuracy of present methods is verified by comparing the consistency of RGT- and GDQM-based normalized linear vibration frequencies. The RGT and GDQM predict almost similar results of nonlinear vibration frequencies in the case of SS microbeams, while GDQM can provide accurate prediction for CC and CS boundary conditions.

Analytical and Numerical Study on Size-Dependent Nonlinear Free Vibration of FG Nonlocal Timoshenko Microbeam in Thermal Environment

期刊论文
发表日期：

2025年11月28日

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Yuan Tang,
Hai Qing

摘要

Abstract

In this paper, the stress-driven \(\left(\sigma D\right)\) and strain-driven \(\left(\varepsilon D\right)\) two-phase local/nonlocal integral models(TPNIM) are employed to formulate the size-dependent nonlinear free vibration of functionally graded (FG) Timoshenko microbeams in a thermal environment. The nonlinear differential governing equations and boundary conditions of motion are derived based on Hamilton’s principle andVon Karman's nonlinearity. The integral constitutive equations between strain and nonlocal stress components are transformed unitedly into equivalent differential forms with constitutive constraints. Neglecting the nonlinear items, the linear vibration frequencies and corresponding linear vibration mode shapes (LVMS) can be determined through the Laplace transformation technique. Based on LVMS, the nonlinear free vibration frequency is expressed explicitly as a function of vibration amplitude through the Ritz-Galerkin technique (RGT). Meanwhile, the numerical method based on the generalized differential quadrature method (GDQM) together with Newton’s iterative process is directly employed to calculate the nonlinear free vibration frequencies for different vibration amplitude. The effects of the relevant parameters on the linear and nonlinear vibration frequencies are investigated theoretically and numerically for microbeams.

Purpose

In this paper, the stress-driven and strain-driven two-phase local/nonlocal integral models(TPNIM) are employed to formulate the size-dependent nonlinear free vibration of functionally graded (FG) Timoshenko microbeams in a thermal environment.

Methods

The nonlinear differential governing equations and boundary conditions of motion are derived based on Hamilton’s principle and Von Karman's nonlinearity. The integral constitutive equations between strain and nonlocal stress components are transformed unitedly into equivalent differential forms with constitutive constraints. Neglecting the nonlinear items, the linear vibration frequencies and corresponding linear vibration mode shapes (LVMS) can be determined through the Laplace transformation technique. Based on LVMS, the nonlinear free vibration frequency is expressed explicitly as a function of vibration amplitude through the Ritz-Galerkin technique (RGT). Meanwhile, the numerical method based on the generalized differential quadrature method (GDQM) together with Newton’s iterative process is directly employed to calculate the nonlinear free vibration frequencies for different vibration amplitude.

Results

The effects of the FG-index, environmental temperature variation and nonlocal length-scale parameters on the linear and nonlinear vibration frequencies are investigated theoretically and numerically for microbeams under different boundary conditions.

Conclusion

The accuracy of present methods is verified by comparing the consistency of RGT- and GDQM-based normalized linear vibration frequencies. The RGT and GDQM predict almost similar results of nonlinear vibration frequencies in the case of SS microbeams, while GDQM can provide accurate prediction for CC and CS boundary conditions.