International Journal of Non-Linear Mechanics最新文献_第9页

Soliton inspired hybrid active vibration control method for shock-induced transient vibrations: Numerical perspective 冲击瞬态振动的孤子激励混合主动振动控制方法：数值视角

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-06-26 DOI: 10.1016/j.ijnonlinmec.2025.105198

Serkan Guler

A novel hybrid control approach has been designed with the primary objective of minimizing vibration amplitude as quickly and effectively as possible, particularly in response to mechanical shocks. The control approach taken in this study is a hybrid methodology based on the integration of the soliton-inspired wave and the conventional Proportional-Integral-Derivative (PID) control technique. The proposed hybrid control approach is compared with the conventional PID control technique considering two-DOF, four-DOF and five-DOF vibratory systems. Mathematical models of these vibratory systems are established and simulated numerically. In these simulations, it was found that the presented control technique further mitigates vibrations than conventional PID. Moreover, to investigate the applicability of the hybrid control methodology in this research to real engineering systems, a half-vehicle model and bolted cantilever beams are considered. Numerical simulation studies are carried out for these two different engineering systems. Through numerical simulation studies conducted on these systems, it was revealed that the proposed hybrid control approach exhibits superior active vibration control performance to the conventional PID method.

设计了一种新的混合控制方法，其主要目标是尽可能快速有效地减小振动幅度，特别是在响应机械冲击时。本研究采用的控制方法是基于孤子启发波和传统的比例积分导数（PID）控制技术的集成的混合方法。针对二自由度、四自由度和五自由度振动系统，将所提出的混合控制方法与传统PID控制方法进行了比较。建立了振动系统的数学模型，并进行了数值模拟。在这些仿真中，发现所提出的控制技术比传统的PID进一步减轻了振动。此外，为了研究本研究中混合控制方法在实际工程系统中的适用性，考虑了半车模型和螺栓悬臂梁。对这两种不同的工程系统进行了数值模拟研究。通过对这些系统的数值仿真研究表明，所提出的混合控制方法比传统PID方法具有更好的主动振动控制性能。

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引用次数: 0

Load force effect on the dynamic characteristics of planar multibody systems with clearances 载荷力对带间隙平面多体系统动力学特性的影响

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-07-03 DOI: 10.1016/j.ijnonlinmec.2025.105202

Xin Fang , Jianghao Wu , Feng Du

Mechanism bears various types of loads. However, the effect of load type and magnitude on the dynamic characteristics of mechanism is less clear. This study investigates the effect of load force on the dynamic characteristics of planar mechanisms with clearance through theoretical modeling and experimental method. The dynamic model of mechanism with clearance is formulated utilizing the Lagrange multiplier method, incorporating a contact force model based on the Lankarani-Nikravesh and modified Coulomb models. Dynamic analysis of a slider-crank mechanism is conducted, revealing a transition from chaotic to periodic behavior with increasing force magnitude. The findings demonstrate the universal behavior across different load force forms, parametric conditions and topological configurations. Experimental validation is conducted on a multi-link mechanism with clearance joints and shows good agreement with theoretical prediction. A comparative analysis between a ball-plane impact model and the slider-crank mechanism with clearance is performed and the effect of load type and magnitude is analyzed respectively. This finding highlights the critical role of load force in determining the dynamic characteristics of planar multibody systems with clearances, offering valuable insights for designing mechanical systems under diverse load conditions.

机构承受各种类型的载荷。然而，载荷类型和大小对机构动力特性的影响尚不清楚。通过理论建模和实验方法，研究了载荷力对带间隙平面机构动态特性的影响。结合基于Lankarani-Nikravesh模型和修正的Coulomb模型的接触力模型，利用拉格朗日乘数法建立了带间隙机构的动力学模型。对曲柄滑块机构进行了动力学分析，揭示了该机构随着力的增大由混沌行为向周期行为转变。研究结果表明，在不同的载荷、力形式、参数条件和拓扑结构中，结构具有普遍的特性。对带间隙连杆的多连杆机构进行了实验验证，结果与理论预测吻合较好。将球面冲击模型与带间隙的曲柄滑块机构进行了对比分析，分别分析了载荷类型和载荷大小对冲击模型的影响。这一发现强调了载荷力在确定带间隙平面多体系统动态特性中的关键作用，为设计不同载荷条件下的机械系统提供了有价值的见解。

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引用次数: 0

Nonlinear buckling analysis of helicoidal composite panels with curvilinear stiffeners 曲线加筋螺旋复合材料板非线性屈曲分析

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-06-25 DOI: 10.1016/j.ijnonlinmec.2025.105195

Xin Lian , Haotian Wei , Weidong Zhang , Yuming Mao , Dongjie Jiang , Zhefeng Yu

This study introduces a semi-analytical method designed to efficiently and accurately analyze the nonlinear buckling response of complex curvilinear stiffened panel structures, addressing the computational challenges associated with such designs. The proposed method establishes displacement compatibility conditions to couple stiffeners and base plate, while employing Legendre polynomials to expand displacement fields, thereby enhancing robustness and accuracy. The Ritz method is utilized to solve the nonlinear buckling equations, yielding equilibrium path deviations consistently below 5 % compared to finite element method results while achieving a 61.43 % reduction in computation time. Parametric studies conducted under uniaxial and biaxial compressive loads confirm the capability of the method to accurately capture the nonlinear buckling behavior of curvilinear stiffened panels. The findings reveal a reduction in nonlinear critical loads, approximately 20 % lower than those predicted by linear buckling analysis, emphasizing the necessity of nonlinear analysis for accurate system evaluation. The study underscores the potential of the proposed semi-analytical method as a reliable and computationally efficient tool for nonlinear buckling analysis in complex stiffened structures.

本研究介绍了一种半解析方法，旨在有效和准确地分析复杂曲线加筋板结构的非线性屈曲响应，解决与此类设计相关的计算挑战。该方法通过建立位移相容条件来耦合加强筋与基板，同时利用勒让德多项式展开位移场，提高了鲁棒性和精度。采用Ritz方法求解非线性屈曲方程，与有限元方法相比，平衡路径偏差始终小于5%，计算时间减少了61.43%。在单轴和双轴压缩载荷下进行的参数化研究证实了该方法能够准确地捕捉曲线加筋板的非线性屈曲行为。研究结果表明，非线性临界载荷的减少，比线性屈曲分析预测的减少了约20%，强调了非线性分析对准确系统评估的必要性。该研究强调了所提出的半解析方法作为复杂加筋结构非线性屈曲分析的可靠且计算效率高的工具的潜力。

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引用次数: 0

Free vibration and large deformation characteristics of metamaterial thickness-deformable plates with initial geometrical imperfection 具有初始几何缺陷的超材料厚度变形板的自由振动和大变形特性

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-06-26 DOI: 10.1016/j.ijnonlinmec.2025.105190

Behrouz Karami , Mergen H. Ghayesh , Shahid Hussain

A geometrically imperfect third-order auxetic metamaterial thickness- and shear-deformable plate is considered, and both the non-linear bending as well as the free vibrations are analysed. Distribution of the effective material properties from plate’s top surface to the bottom one follows a functionally graded form; material properties are effectively approximated via genetic programming-assisted micromechanics models from previous studies. Without ignoring geometrical non-linearities, the fourfold coupled axial, transverse, rotational, and stretching non-linear motion equations are formulated for a geometrically imperfect third-order auxetic metamaterial thickness- and shear-deformable plate using Hamilton’s principle. The generalised differential quadrature method is implemented to discretise the motion equations; the discretised motion equations are then solved for both the non-linear bending as well as the linear free vibrations. For partial validation, the model is compared with available data from the open literature for simplified cases (i.e., single-layer homogeneous plates without metamaterial characteristics) and with a single-layered isotropic rectangular perfectly straight plate modelled in ANSYS. The complex non-linear mechanics and linear free vibration of the metamaterial system are analysed for different geometrical parameters, graphene origami contents, folding degrees, and geometrical imperfections, and also for both the symmetric and asymmetric distribution patterns of graphene origami. The results reveal that the geometric imperfections reduce the transverse deflection, and metamaterial plates with asymmetric graphene origami distributions consistently have the largest non-linear deflections among all the graphene origami distributions studied.

考虑了一种几何不完美的三阶形变超材料厚度和剪切变形板，对其非线性弯曲和自由振动进行了分析。材料的有效性能从板的上表面到板的下表面呈功能梯度分布；通过遗传规划辅助的微观力学模型可以有效地逼近材料的性能。在不忽略几何非线性的前提下，利用Hamilton原理建立了几何不完美三阶形变超材料厚度和剪切变形板的轴向、横向、旋转和拉伸四重耦合非线性运动方程。采用广义微分正交法对运动方程进行离散化；然后求解了非线性弯曲和线性自由振动的离散运动方程。为了进行部分验证，将该模型与公开文献中简化情况（即，不具有超材料特征的单层均匀板）的可用数据以及在ANSYS中建模的单层各向同性矩形完美直板进行了比较。分析了不同几何参数、石墨烯折纸含量、折叠程度、几何缺陷以及石墨烯折纸的对称和非对称分布模式下，超材料系统的复杂非线性力学和线性自由振动。结果表明，几何缺陷减少了横向挠度，具有不对称石墨烯折纸分布的超材料板在所有研究的石墨烯折纸分布中始终具有最大的非线性挠度。

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引用次数: 0

Vibration behavior of laminated couplings based on angular misalignment and nonlinear stiffness characteristics 基于角不对中和非线性刚度特性的层合联轴器振动特性

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-05-27 DOI: 10.1016/j.ijnonlinmec.2025.105175

Yong Huang, Fei Peng

The coupled bending-torsion vibration model of the laminated coupling was established, taking into account nonlinear stiffness, mass eccentricity, and angular misalignment. First, a laminated coupling vibration test study was conducted. The results showed that the average relative error between the numerical model and the test results was 10.2 %. Subsequently, the effects of the misalignment amount and nonlinear stiffness on the vibration characteristics of the laminated coupling were analyzed. The results showed that an increase in the angular misalignment amount leads to an increase in the vibration amplitude of the laminated coupling. Additionally, nonlinear stiffness has a suppressive effect on the vibration amplitude of the system. Finally, the effects of the angular misalignment amount, bending/torsional stiffness critical values, and rotational speed on the vibration behavior of the laminated coupling were observed using bifurcation diagrams. The research results show that when the angular misalignment exceeds 2.3 × 10⁻³ rad, the critical value of bending stiffness is greater than 8.44 × 10⁶ N/m, and the critical value of torsional stiffness is lower than 2.88 × 10⁶ N·m/rad, the vibration amplitude of the laminated coupling tends to rise, and resonance may occur. Additionally, controlling the laminated coupling vibration amplitude is more effective when the rotational speed is either in the low-speed range or the high-speed range. The research results detailed in this paper not only inform the design of laminated coupling parameters but also provide guidance for the assembly process.

建立了考虑非线性刚度、质量偏心和角度偏差的层合联轴器弯扭耦合振动模型。首先，进行了层合耦合振动试验研究。结果表明，数值模型与试验结果的平均相对误差为10.2%。分析了错位量和非线性刚度对层合联轴器振动特性的影响。结果表明，角不对中量的增大导致层合联轴器的振动幅值增大。此外，非线性刚度对系统的振动幅值有抑制作用。最后，利用分岔图分析了角不对中量、弯曲/扭转刚度临界值和转速对层合联轴器振动特性的影响。研究结果表明，当角度偏差超过2.3 × 10−3 rad时，弯曲刚度临界值大于8.44 × 106 N/m，扭转刚度临界值小于2.88 × 106 N·m/rad，层合联轴器的振动幅值趋于上升，并可能发生共振。此外，在低速和高速转速范围内，层合耦合振动幅值控制更为有效。本文的研究结果不仅为层合耦合参数的设计提供了依据，也为装配过程提供了指导。

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引用次数: 0

Infrasound amplification using simple strongly nonlinear granular metamaterials 利用简单的强非线性颗粒超材料进行次声放大

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-06-02 DOI: 10.1016/j.ijnonlinmec.2025.105160

Elbens Carlos Viana Reis , Luis Paulo Silveira Machado , Surajit Sen

Monodispersed and inverse tapered granular chains with grain sizes varying uniformly from small to large under external harmonic excitations are numerically investigated. The control parameters are the number of granules, driving amplitude, driving frequency, and tapering parameter. The frequency response is analyzed using the transfer function, where properties of transmission, amplification, and filtering of signals are observed. For monodispersed chains, amplification peaks are described by functions of the control parameters. For sufficiently long monodispersed chains, signals at low frequencies (

\approx 10 Hz

) are amplified. Inverse tapered chains exhibit similar behavior with shorter chain lengths, as do monodispersed chains modeled with forces between granules with nonlinearity greater than that of the Hertzian contact law. Potential applications are in the fields of geosciences and oceanography, life sciences, military applications, and environmental monitoring.

本文对外部谐波激励下粒径由小到大均匀变化的单分散和反锥形颗粒链进行了数值研究。控制参数为颗粒数、驱动振幅、驱动频率和锥度参数。使用传递函数分析频率响应，其中观察信号的传输、放大和滤波特性。对于单分散链，放大峰由控制参数的函数来描述。对于足够长的单分散链，低频（≈10Hz）的信号被放大。反锥形链在更短的链长下表现出类似的行为，单分散链在非线性大于赫兹接触律的颗粒间力模型中也表现出类似的行为。潜在的应用领域包括地球科学和海洋学、生命科学、军事应用和环境监测。

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引用次数: 0

High order invariant manifold model reduction for systems with non-polynomial non-linearities: Geometrically exact finite element structures and validity limit 非多项式非线性系统的高阶不变流形模型约简：几何精确有限元结构与有效性限制

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-05-31 DOI: 10.1016/j.ijnonlinmec.2025.105138

Aurélien Grolet , Alessandra Vizzaccaro , Marielle Debeurre , Olivier Thomas

This paper considers the computation of reduced-order models for systems of ordinary differential equations that include non-polynomial non-linearities. A targeted example is the case of a geometrically exact model of highly flexible slender structure, that includes, after space discretisation, trigonometric non-linear terms. With a suitable change of variables, this system can be rewritten in an equivalent one with polynomial non-linearities at most quadratic, at the price of introducing additional variables linked to algebraic equations, leading to a differential algebraic set of equations (DAE) to be solved. This DAE is reduced thanks to a normal form parametrisation of its invariant manifolds and selecting a set of master ones. Arbitrary order expansions are detailed for the coefficients of the change of variable and the reduced dynamics, using linear algebra in the space of multivariate polynomials of a given degree. In the case of a single non-linear mode reduction, a criterion to evaluate the quality of the normal form results is also proposed based on an estimation of the convergence radius of the polynomial asymptotic expansion representing truncated series. The method is then applied to compute a single mode reduction of three test cases – a Duffing oscillator, a simple pendulum and a clamped clamped beam with von Kármán model –, in order to investigate the effect of the algebraic part of the DAE on the quality of the model reduction and its validity range. Then, the more involved case of a cantilever beam modelled by geometrically exact finite elements is considered, underlining the ability of the method to produce accurate and converged results in a range of amplitude that can be bounded thanks to a convergence criterion.

研究含非多项式非线性的常微分方程系统的降阶模型的计算。一个有针对性的例子是高度灵活的细长结构的几何精确模型，其中包括经过空间离散化的三角非线性项。通过适当的变量变换，该系统可以被重写为一个多项式非线性的等效系统，其代价是引入与代数方程相关的额外变量，从而导致求解微分代数方程组（DAE）。由于对其不变流形进行了正规的参数化，并选择了一组主流形，因此减小了DAE。在给定次数的多元多项式空间中，利用线性代数详细地对变量变化系数和简化动力学进行了任意阶展开。在单非线性模态约简的情况下，基于截断级数的多项式渐近展开式的收敛半径估计，提出了一个评价范式结果质量的准则。将该方法应用于Duffing振子、单摆和箝位梁三个测试用例的von Kármán模型的单模约简计算，以探讨DAE的代数部分对模型约简质量及其有效范围的影响。然后，考虑了更复杂的由几何精确有限元建模的悬臂梁的情况，强调了该方法在可以由于收敛准则而有界的幅度范围内产生准确和收敛结果的能力。

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引用次数: 0

Model reduction of a flexible nonsmooth oscillator recovers its entire bifurcation structure 柔性非光滑振子的模型简化恢复了其整个分岔结构

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-07-02 DOI: 10.1016/j.ijnonlinmec.2025.105194

Suparno Bhattacharyya, Joseph P. Cusumano

We study the reduced order modeling of a piecewise-linear, globally nonlinear flexible oscillator in which a Bernoulli–Euler beam is subjected to a position-triggered kick force and a piecewise restoring force at its tip. The nonsmooth boundary conditions, which determine different regions of a hybrid phase space, can generally be expected to excite many degrees of freedom. With kick strength as parameter, the system’s bifurcation diagram is found to exhibit a range of periodic and chaotic behaviors. Proper orthogonal decomposition (POD) is used to obtain a single set of global basis functions spanning all of the hybrid regions. The reduced order model (ROM) dimension is chosen using previously developed energy closure analysis, ensuring approximate energy balance on the reduced subspace. This yields accurate ROMs with 8 degrees of freedom. Remarkably, we find that ROMs formulated using data from individual periodic steady states can nevertheless be used to reconstruct the entire bifurcation structure of the original system without updating. This demonstrates that, despite being constructed with steady state data, the ROMs model sufficiently small transients with enough accuracy to permit using simple continuation for the bifurcation diagram. We also find ROM subspaces obtained for different values of the bifurcation parameter are essentially identical. Thus, POD augmented with energy closure analysis is found to reliably yield effective dimension estimates and ROMs for this nonlinear, nonsmooth system that are robust across stability transitions, including even period doubling cascades to chaos, thereby greatly reducing data requirements and computational costs.

本文研究了分段线性全局非线性挠性振荡器的降阶建模，其中伯努利-欧拉梁受到位置触发的踢力和末端的分段恢复力的作用。非光滑边界条件决定了混合相空间的不同区域，通常可以期望激发多个自由度。以井涌强度为参数，发现系统的分岔图表现出一系列周期和混沌行为。采用适当的正交分解（POD）方法得到一组跨越所有混合区域的全局基函数。使用先前开发的能量闭合分析选择降阶模型（ROM）维度，确保约简子空间上的近似能量平衡。这产生具有8个自由度的精确rom。值得注意的是，我们发现使用单个周期稳态数据制定的rom仍然可以用来重建原始系统的整个分岔结构而无需更新。这表明，尽管是用稳态数据构建的，但ROMs模型足够小的瞬态，具有足够的精度，可以使用简单的分岔图延拓。我们还发现对于不同的分岔参数值所得到的ROM子空间本质上是相同的。因此，我们发现，结合能量闭合分析的POD可以可靠地为这种非线性、非光滑系统产生有效的尺寸估计和rom，这些系统在稳定性过渡（甚至包括周期加倍级联到混沌）中都具有鲁棒性，从而大大降低了数据需求和计算成本。

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引用次数: 0

Uncertainty propagation analysis in a double-disk rotor system using a kriging surrogate model and ellipsoidal convex set boundaries 基于kriging代理模型和椭球凸集边界的双盘转子系统不确定性传播分析

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-05-09 DOI: 10.1016/j.ijnonlinmec.2025.105150

Yunlong Zhao , Ruize Gao , Wangqun Deng , Mingming Shi , Li Huang , Wenkui Liu , Yongfeng Yang

This study analyzes the sensitivity of a double-disk rotor system to uncertain parameters and its nonlinear responses, while also conducting a propagation analysis for both single and multiple uncertain parameters. In this study, a dynamic model of a simply supported double-disk rotor system is developed using the finite element method. Additionally, the ellipsoidal convex set model is employed to quantify the uncertainty parameters in the rotor system. A Kriging surrogate model is developed to efficiently and accurately predict the nonlinear response of the rotor system under uncertainty. The global sensitivity of the rotor system's vibration response is analyzed using the Sobol global sensitivity index. Parameters with higher sensitivity indices are subsequently incorporated into the surrogate model to investigate the propagation mechanisms of both single and multiple uncertainty parameters of the rotor system. The results indicate that the critical speed of the rotor system is primarily influenced by the rotor density and elastic modulus. The vibration peaks of the rotor system are primarily influenced by the disk mass, viscous damping, and the degree of unevenness. Under the influence of multi-parameter uncertainty, the fluctuation ranges of the rotor system's peak vibration and critical speed are further extended.

本文分析了双盘转子系统对不确定参数的敏感性及其非线性响应，并对单不确定参数和多个不确定参数进行了传播分析。本文采用有限元方法建立了简支双盘转子系统的动力学模型。此外，采用椭球凸集模型对转子系统中的不确定性参数进行量化。为了有效准确地预测转子系统在不确定条件下的非线性响应，建立了Kriging代理模型。利用Sobol全局灵敏度指数分析了转子系统振动响应的全局灵敏度。随后将灵敏度指数较高的参数纳入代理模型，研究转子系统单不确定参数和多不确定参数的传播机理。结果表明，转子系统的临界转速主要受转子密度和弹性模量的影响。转子系统的振动峰值主要受圆盘质量、粘性阻尼和不均匀度的影响。在多参数不确定性的影响下，进一步扩大了转子系统峰值振动和临界转速的波动范围。

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引用次数: 0

Geometrically nonlinear oscillations of composite sandwich cylindrical shell with honeycomb core under axial time periodic force 轴向时间周期力作用下蜂窝芯复合材料夹层圆柱壳的几何非线性振动

IF 2.8 3区工程技术 Q2 MECHANICS

International Journal of Non-Linear Mechanics

Pub Date : 2025-11-01 Epub Date: 2025-06-25 DOI: 10.1016/j.ijnonlinmec.2025.105196

B. Uspensky , K. Avramov , S. Malyshev , O. Nikonov

Cylindrical composite sandwich shell, which consists of two outer layers and thick honeycomb core, is considered. The outer thin layers are manufactured from composite orthotropic material and honeycomb core is manufactured from orthotropic plastic.

Parametric nonlinear oscillations of cylindrical shell under the action longitudinal time periodic force are considered. The honeycomb core is homogenized. As a result, orthotropic solid continuum is obtained. Stressed state of every layer is described by higher order shear theory, which uses five generalized displacements (three displacements projections and two rotations angles of normal to middle surfaces). The assumed-mode method is applied to obtain the system of nonlinear ordinary differential equations with respect to the generalized coordinates to describe the sandwich structure vibrations.

The shooting technique and continuation method are applied jointly to analyze the nonlinear oscillations, their stability and bifurcations. The geometrically nonlinear oscillations are considered in the principal parametric resonances with account of internal resonances. Stability and bifurcations of periodic motions are shown on the frequency response, which describes the structure nonlinear dynamics in principle parametric resonances.

研究了由两层外层和厚蜂窝芯组成的圆柱形复合夹层壳。外层薄层由复合正交异性材料制成，蜂窝芯由正交异性塑料制成。考虑了圆柱壳在纵向时间周期力作用下的参数非线性振动。蜂窝芯均匀化。得到正交各向异性固体连续体。每层的受力状态用高阶剪切理论来描述，该理论使用5个广义位移（3个位移投影和2个法向-中间面的旋转角）。采用假设模态法，得到了在广义坐标下描述夹层结构振动的非线性常微分方程组。采用射击法和延拓法对非线性振动、稳定性和分岔进行了分析。在考虑内共振的情况下，考虑了主参数共振的几何非线性振荡。频率响应表现出周期运动的稳定性和分岔性，从原理上描述了结构的非线性动力学参数共振。

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引用次数: 0