Abstract Multi-step Timoshenko beams coupled with rigid bodies on springs can be regarded as a generalized model to investigate the dynamic characteristics of many structures and mechanical systems in engineering. This paper presents a novel transfer matrix method for the free and forced vibration analyses of the hybrid system. It is modeled as a chain system, where each beam and each rigid body with its supporting spring are dealt with one element, respectively. The transfer equation of each element is deduced based on separation of variables method. The system overall transfer equation is obtained by substituting an element transfer equation into another. Then, the free vibration characteristics are acquired by solving exact homogeneous linear equations. To compute the forced vibration response with modal superposition method, the body dynamic equations and augmented eigenvectors are established, and the orthogonality of augmented eigenvectors is mathematically proved. Without high-order global dynamic equation or approximate spatial discretization, the free and forced vibration analyses of the hybrid system are achieved efficiently and accurately in this study. As an analytical approach, the present method is easy, highly stylized, robust, powerful and general for the complex hybrid systems containing any number of Timoshenko beams and rigid bodies. Four numerical examples are implemented, and the results show that this method is computationally efficient with high precision.

中文翻译：

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多阶铁木辛哥梁与弹簧上刚体耦合的自由和受迫振动分析的传递矩阵方法

摘要 多阶铁木辛柯梁与弹簧上的刚体耦合可作为研究工程中许多结构和机械系统动力学特性的广义模型。本文提出了一种新的传递矩阵方法，用于混合动力系统的自由振动和受迫振动分析。它被建模为一个链系统，其中每个梁和每个刚体及其支撑弹簧分别处理一个元素。基于变量分离法推导出各单元的传递方程。系统整体传递方程是将一个元素传递方程代入另一个方程。然后，通过求解精确的齐次线性方程来获得自由振动特性。用模态叠加法计算受迫振动响应，建立了体动力学方程和增广特征向量，并从数学上证明了增广特征向量的正交性。在没有高阶全局动力学方程或近似空间离散化的情况下，本研究高效准确地实现了混合系统的自由和受迫振动分析。作为一种分析方法，本方法对于包含任意数量 Timoshenko 梁和刚体的复杂混合系统来说是简单、高度程式化、稳健、强大和通用的。实施了四个数值例子，结果表明该方法计算效率高，精度高。在没有高阶全局动力学方程或近似空间离散化的情况下，本研究高效准确地实现了混合系统的自由和受迫振动分析。作为一种分析方法，本方法对于包含任意数量 Timoshenko 梁和刚体的复杂混合系统来说是简单、高度程式化、稳健、强大和通用的。实施了四个数值例子，结果表明该方法计算效率高，精度高。在没有高阶全局动力学方程或近似空间离散化的情况下，本研究高效准确地实现了混合系统的自由和受迫振动分析。作为一种分析方法，本方法对于包含任意数量 Timoshenko 梁和刚体的复杂混合系统来说是简单、高度程式化、稳健、强大和通用的。实施了四个数值例子，结果表明该方法计算效率高，精度高。