摘要：无论是在自然界、机械技术、工程应用还是社会日常生活中，事物状态的循环变化以及物体的往复运动都极为广泛的存在着，而这类现象通常统称为振荡。振动是振荡中一种较为特殊的情况，一般是指物体在平衡位置附近较为微小或者说有限的振荡。机械振动是指工程机械技术中涉及的有关机械器件和建筑系统结构的振动。振动力学是研究机械振动的运动学和动力学的一门课程。在力作用下发生了弯曲变形的杆件即称为梁，梁的振动是振动力学中一种较为特殊的振动。本课题要求建立两端弹性约束梁的微分方程模型，并求解相应的特征值问题。主要内容：对梁的力学模型的类别进行简要阐述，分析影响梁的振动及弯曲的各类因素，利用常微分及振动力学知识建立两端弹性约束梁的约束模型，求解模型的特征值，并进行应用说明。70113

毕业论文关键词：振动，梁，两端弹性约束，特征值，偏微分方程 

The eigenvalue problem of the elastic constraint beam on both ends

Abstract: Both, mechanical technology and engineering application in nature and society in daily life, the circulation of the state of things change and the reciprocating movement of the object are widely exist, and this kind of phenomenon is often referred to as oscillation. Vibration is a special case of oscillation, usually a small or limited oscillation near the equilibrium position. Mechanical vibration refers to the vibration of mechanical device and building system structure involved in engineering mechanical technology. Vibration mechanics is a course in kinematics and dynamics of mechanical vibration. Under the influence of force, the bending deformation of the bar is called the beam, and the vibration of the beam is a kind of special vibration in vibration mechanics. In this paper, the differential equation model of elastic constrained beam is established, and the corresponding eigenvalue problem is solved. Main contents: the beam mechanical model, the article briefly expounds the category of the analysis of various factors affecting the vibration of the beam and the bending, by using differential and vibration mechanics knowledge both ends of a beam on a  

elastic constraints constraint model is established, to solve the eigenvalue of the model, and application instructions.

Keywords: vibration, beam, both sides elastic constraints, eigenvalues, partial differential equations

目录

摘要 i

Abstract i

1.绪论 1

   1.1关于振动 1

1.2 梁及梁的弯曲振动 1

2.模型介绍 2

   2.1梁弯曲振动的力学模型 2

   2.2两端弹性约束梁 2

3.梁弯曲振动的公式推导 3

   3.1 运动微分方程 3

   3.2梁弯曲振动的解 5

3.3 各类梁的边界条件 6

3.4 等截面梁的弯曲振动 8

4. 两端弹性约束梁的特征值问题 9

   4.1 两端弹性支座梁 9

   4.2两端弹性固定梁 11

   4.3其他梁的应用延伸 12

5. 结论