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Keywords: Nonlinear free vibration; Portable accelerometer; Experimental and analytical methods; FSDT; Multiple-scale method
关键词：非线性自由振动;便携式加速度计;实验和分析方法;FSDT;多尺度方法

In most engineering applications, the structure behavior has been considered linear due to the linear material properties, while in some real-world applications, the structures deal with large deformation which causes nonlinear behavior. The understanding of initial conditions and their effect on the nonlinear free vibrations of structures is of paramount importance in engineering and structural dynamics. An indepth investigation to reveal the relationship between initial conditions and the resulting vibrational behavior is crucial. By studying the effects of various initial conditions, such as displacement amplitudes, we aim to enhance our comprehension of the dynamic response of thin-walled beams, ultimately contributing to the design and optimization of these vital engineering components. Through analysis and experimentation, we endeavor to shed light on the hardening and softening phenomenon in nonlinear free vibrations and present the difference between linear and nonlinear frequencies.
在大多数工程应用中，由于线性材料属性，结构行为被认为是线性的，而在某些实际应用中，结构会处理导致非线性行为的大变形。了解初始条件及其对结构非线性自由振动的影响在工程和结构动力学中至关重要。深入调查以揭示初始条件与产生的振动行为之间的关系至关重要。通过研究各种初始条件（如位移幅值）的影响，我们的目标是提高对薄壁梁动态响应的理解，最终为这些重要工程组件的设计和优化做出贡献。通过分析和实验，我们努力阐明非线性自由振动中的硬化和软化现象，并提出线性频率和非线性频率之间的差异。

Duan (1) presented a finite element (FE) model to study the nonlinear free vibration of thin-walled beams with asymmetric open cross-sections by employing cubic polynomials as the shape functions. Ke et al. (2) studied the nonlinear vibrations of functionally graded (FG) Euler-Bernoulli (EB) beams using the von Karman relation. The solution was proposed using the Galerkin procedure and Runge-Kutta method. Sedighi et al. (3) presented an innovative equivalent function to analyze the frequency-amplitude relation
段 （1） 提出了一个有限元 （FE） 模型，通过采用三次多项式作为形函数来研究具有不对称开放截面的薄壁梁的非线性自由振动。Ke 等人 （2） 使用 von Karman 关系研究了功能梯度 （FG） 欧拉-伯努利 （EB） 光束的非线性振动。该解决方案是使用 Galerkin 程序和 Runge-Kutta 方法提出的。Sedighi 等人 （3） 提出了一种创新的等效函数来分析频率-振幅关系

of nonlinear beam vibrations. Gonzalez-Cruz et al. (4) studied large deformation effects on nonlinear vibrations of cantilever beams using an experimental procedure and continuous wavelet transform. Chen et al. (5) investigated the nonlinear free vibrations of sandwich porous Timoshenko beams based on the von Karman relations and Ritz method. The linear and nonlinear frequencies were determined by conducting a direct iterative algorithm. Ghasemi and Mohandes (6) used finite strain assumptions and generalized differential quadrature (DQ) method to study the free vibration of laminated composite EB beams. Panigrahi and Pohit (7) studied the effect of stiffening due to rotations on the nonlinear vibrations of cracked Timoshenko FG beams. The governing equations were solved using the Ritz method and direct substitution iterative technique. Arani et al. (8) investigated free and forced vibration of EB sandwich nano-beams with different FG patterns of reinforced nanotubes face sheets using von Karman relations. The solution was proposed based on the Galerkin method and the multiple-scale technique. Rajabi et al. (9) examined nonlinear vibration response of EB nanobeams under harmonic moving load by employing the Galerkin method and multistage-linearization technique. A theoretical study of stability and nonlinear dynamic behavior of a nano beam under an axially time-dependent load was presented by Chen et al. (10). The Floquet technology and Bolotin’s method were employed to linearly study the stability of the structure. Shafei et al. (11) examined geometrically nonlinear vibration of composite beams based on the third-order of shear deformation theory (TSDT), von Karman relations, and Newton-Raphson method. Alambeigi et al. (12) investigated the free and forced vibration of a sandwich FG beam with a porous core. The equations were derived based on the FSDT and they were solved using the Navier solution. Ghuku et al. (13) presented an experimental and theoretical study on the free vibration of statically loaded moving boundary beam by considering shear deformation effects. Fontanela et al. (14) reported nonlinear vibration localization of two coupled beams by employing a two-degree-of-freedom model with piecewise linear stiffness. Kheladi et al. (15) used a new isogeometric beam element to study the free vibrations of composite Timoshenko beams based on the equivalent single layer theory. Magnucki et al. (16) determined free flexural vibration response of beams with variable cross-section by employing Hamilton’s principle. Sohani and Eipakchi (17) used the multiple-scale method to solve nonlinear equations of motion of beam with nonuniform cross-section based on the FSDT and von Karman relations. Kharazan et al. (18) studied the nonlinear response of cantilever beam with transverse crack and bilinear behavior using the polynomial functions. Elkaimbillah et al. (19) presented a one-dimensional FE model to determine the nonlinear dynamic behavior of thin-walled composite beams with open crosssections. A free vibration analysis of an auxetic FG Timoshenko beam was conducted by Zhao et al. (20) based on the von Karman relations and by employing DQ method. Eipakchi and Mahboubi Nasrekani (21) studied the free vibration of composite beams with auxetic honeycomb core layer based on the FSDT and von Karman relations. Lin et al. (22) presented an investigation into the effects of cone ratio and axial loading on the vibrational behavior of non-uniform cantilever EB beams using the Laplace Adomian decomposition method. Ghuku et al. (23) studied the forced vibrations of curved beams under static load and harmonic excitation. The solution was proposed using the Broyden method. Sohani and Eipakchi (24) investigated the nonlinear geometry effects on the free vibrations of beams based on the FSDT and von Karman relations and the multiple-scale method. Esen et al. (25) studied the forced vibrations of beams reinforced with carbon nanotubes under moving load using the TSDT and Navier’s method. Nonlinear vibrations of multilayer FG graphene composite Timoshenko microbeams were studied by Zhang et al. (26) by employing the modified couple stress theory and von Karman relations. The numerical solution was determined using the direct iteration technique. Babar and Sahoo (27) investigated the bending, buckling, and free vibrations of composite beams based on inverse hyperbolic shear deformation theory. The governing equations were solved by employing the Navier solution technique. Liu et al. (28) explored the vibration reduction efficiency of Timoshenko beam coupled with a nonlinear energy sink cell element at high excitation frequency. The governing equations of motion were solved using the
非线性梁振动。Gonzalez-Cruz 等人 （4） 使用实验程序和连续小波变换研究了对悬臂梁非线性振动的大变形影响。Chen 等人 （5） 基于 von Karman 关系和 Ritz 方法研究了夹层多孔 Timoshenko 梁的非线性自由振动。线性和非线性频率是通过执行直接迭代算法来确定的。Ghasemi 和 Mohandes （6） 使用有限应变假设和广义差分正交 （DQ） 方法研究了层压复合材料 EB 梁的自由振动。Panigrahi 和 Pohit （7） 研究了旋转引起的刚度对开裂的 Timoshenko FG 梁的非线性振动的影响。使用 Ritz 方法和直接替换迭代技术求解控制方程。Arani 等人 （8） 使用 von Karman 关系研究了增强纳米管面板具有不同 FG 图案的 EB 夹层纳米束的自由和强迫振动。该解决方案基于 Galerkin 方法和多尺度技术提出。Rajabi 等人 （9） 通过采用 Galerkin 方法和多级线性化技术研究了 EB 纳米束在谐波移动载荷下的非线性振动响应。Chen 等人对纳米光束在轴向时间相关载荷下的稳定性和非线性动态行为进行了理论研究 （10）。采用 Floquet 技术和 Bolotin 方法对结构的稳定性进行线性研究。Shafei 等人 （11） 基于三阶剪切变形理论 （TSDT）、von Karman 关系和 Newton-Raphson 方法研究了复合梁的几何非线性振动。Alambeigi 等人。 （12） 研究了具有多孔芯的夹层 FG 梁的自由和强迫振动。这些方程是基于 FSDT 推导出来的，并使用 Navier 解求解。Ghuku 等人 （13） 通过考虑剪切变形效应，对静载荷移动边界梁的自由振动进行了实验和理论研究。Fontanela 等人 （14） 通过采用具有分段线性刚度的二自由度模型，报道了两个耦合梁的非线性振动定位。Kheladi 等人 （15） 使用一种新的等几何梁单元来研究基于等效单层理论的复合 Timoshenko 梁的自由振动。Magnucki 等人 （16） 通过采用汉密尔顿原理确定了具有可变横截面的梁的自由弯曲振动响应。Sohani 和 Eipakchi （17） 使用多尺度方法求解基于 FSDT 和 von Karman 关系的横截面不均匀的梁的非线性运动方程。Kharazan 等人 （18） 使用多项式函数研究了悬臂梁与横向裂纹和双线性行为的非线性响应。Elkaimbillah 等人 （19） 提出了一个一维有限元模型，用于确定具有开放横截面的薄壁复合梁的非线性动力学行为。Zhao 等人 （20） 基于 von Karman 关系并采用 DQ 方法对 auxetic FG Timoshenko 梁进行了自由振动分析。Eipakchi 和 Mahboubi Nasrekani （21） 基于 FSDT 和 von Karman 关系研究了具有蜂窝芯层的复合梁的自由振动。Lin 等人。 （22） 使用拉普拉斯 Adomian 分解方法研究了锥比和轴向载荷对非均匀悬臂 EB 梁振动行为的影响。Ghuku 等人 （23） 研究了静载荷和谐波激励下弯曲梁的强迫振动。该解决方案是使用 Broyden 方法提出的。Sohani 和 Eipakchi （24） 基于 FSDT 和 von Karman 关系以及多尺度方法研究了非线性几何对梁自由振动的影响。Esen 等人 （25） 使用 TSDT 和 Navier 方法研究了碳纳米管增强梁在移动载荷下的强迫振动。Zhang 等人 （26） 采用改进的耦合应力理论和 von Karman 关系研究了多层 FG 石墨烯复合材料 Timoshenko 微束的非线性振动。使用直接迭代技术确定数值解。Babar 和 Sahoo （27） 基于逆双曲剪切变形理论研究了复合梁的弯曲、屈曲和自由振动。控制方程是通过采用 Navier 求解技术求解的。Liu 等人 （28） 探讨了 Timoshenko 束与非线性能量汇单元元件在高激励频率下的减振效率。运动控制方程使用