兰州大学机构库 >土木工程与力学学院
功率流理论及其在杆梁类结构振动分析中的应用
Alternative TitleThe Theory of Power Flow and Its Applications in Vibration Analysis of Rod and Beam Structures
华超锋
Thesis Advisor郭永强
2018-03-21
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword功率流 高阶杆梁理论 回传射线矩阵法 功率流频谱曲线 振动分析
Abstract

与结构动力学传统方法不同,功率流理论是从能量的角度对结构进行动力分析。杆梁类结构不仅可以作为独立结构存在,而且是组成平面或空间框架、网架结构的重要构件,应用功率流理论对其进行振动分析具有极其重要的意义。本文对功率流的理论基础作了系统论述,并将其应用于杆梁类结构的振动分析中,主要内容包括:

(1)以弹性介质构成的一般系统为基本模型,通过物理方程、几何方程和运动方程推导了功率流密度以及瞬态功率流的表达式,进而应用能量定理得到稳态功率流。针对重要的简谐情形,推导了复功率流的表达式,并引入机械阻抗/导纳的概念,导出了其与复功率流之间的关系。(2)分别以Love杆与经典杆理论和Timoshenko梁与Euler-Bernoulli梁理论为理论模型,引入功率流理论,通过杆/梁单元的控制方程推导了杆/梁单元任意截面处的复功率流。以节点处包含集中质量、弹簧及阻尼支承的二相杆/梁结构为模型,引入回传射线矩阵法求解杆/梁结构的波幅解,推导了该杆/梁结构任意截面处复功率流、有功功率流和无功功率流的具体表达式,并以此作为数值算例,分析结构的刚度比,节点集中质量、弹簧刚度、阻尼系数,材料阻尼等因素对功率流的影响。

研究发现:(1)功率流表征结构振动的能量,其中有功功率流对应于行波部分传递的能量,可向远处传播;无功功率流对应于驻波部分相互震荡的能量,不能向远处传播。有功功率流和无功功率流的峰值对应的频率为结构的自振频率。当结构无阻尼时,有功功率流为零,而无功功率流可以反映结构的振动特性。(2)相比于经典杆理论和Euler-Bernoulli梁理论,Love杆理论和Timoshenko梁理论在应用于杆梁结构功率流分析时适用频率更高。(3)对于杆结构,泊松比对低频功率流影响较小,而对高频功率流影响较大。Love杆理论中,纵波传递的能量截止于杆的截止频率,此时纵波不传递能量,能量全部被横向振动吸收。(4)杆梁结构中,结构的刚度比、节点集中质量和弹簧刚度等因素的变化都是通过改变结构的自振频率而对功率流在不同频率范围产生不同的影响。节点集中阻尼支承和材料阻尼的变化不改变结构的自振频率,但其对功率流的峰值(或反向峰值)的大小有影响。随着阻尼系数或材料阻尼因子的增大,功率流峰值减小,在峰值区域外,有功功率流增大,而无功功率流几乎不受影响。

Other Abstract

It is different from the traditional methods of structural dynamics that the theory of power flow is from a point of energy for structure vibration analysis. The rod and beam structures can not only exist as independent structures, but also as important components of the plane or space frame and grid structures, it is of great significance to apply the theory of power flow to vibration analysis on them. In this paper, we systematically discussed the theoretical basis of power flow and applied it to vibration analysis of rod and beam structures, and the main contents include:

(1) The general system composed of elastic media is as the basic model, the expressions of power flow density and transient power flow are derived from the physical equations, geometric equations, and motion equations, then the steady state power flow is obtained from using the energy theorem. For the important harmonic case, the expression of complex power flow is deduced, and then, the concept of mechanical impedance/admittance is introduced, and the relationship between it and complex power flow is derived.(2) Using the theory of Love rod and classical rod, the theory of Timoshenko beam and Euler-Bernoulli beam as the theoretical models, respectively. The complex power flow at any section of the rod/beam element was deduced by the governing equations of the rod/beam element, when the power flow theory was introduced. Taking the two-phase rod/beam structure with concentrated mass lump, spring and damping support at the joint as a model, the method of reverberation-ray matrix (MRRM) was used to solve the wave amplitude solution of the rod/beam structure, and the complex power flow, active power flow and reactive power flow at any section of the rod/beam structure was derived. And then, the structures are used as numerical examples to analyze the influence of the structural stiffness ratio, mass lump, spring stiffness, damping coefficient, and material damping on the power flow.

These results show that: (1) The power flow represents the vibration energy of the structure, in which the active power flow corresponds to the energy transmitted by the traveling wave and can be transmitted to a distant place, and the reactive power flow corresponds to the oscillating energy of the standing wave and cannot be transmitted far away. The frequency corresponding to the peak of active power flow and reactive power flow is the natural frequency of the structure. When the structure is not damped, the active power flow is zero, and the reactive power flow can reflect the vibration characteristics of the structure. (2) Compared to the classical rod theory and the Euler-Bernoulli beam theory, the Love rod theory and the Timoshenko beam theory are more applicable to the higher frequency of power flow analysis of rod and beam structures. (3) For the rod structure, the Poisson's ratio has less influence on the low-frequency power flow, but has a greater impact on the high-frequency power flow. In the Love rod theory, the energy transmitted by the longitudinal wave is cut off from the cutoff frequency of the rod. At this time, the longitudinal wave does not transmit energy, and all the energy is absorbed by the transverse vibration. (4) For the rod and beam structure, the structural stiffness ratio, the concentrated mass lump and the spring stiffness of joint and other factors all have different effects on the power flow in different frequency ranges by changing the natural frequency of the structure. The change of the joint damping and material damping does not change the natural frequency of the structure, but it has an influence on the magnitude of the peak (or reverse peak) of the power flow. With the increase of joint damping coefficient or material damping factor, the power flow peak decreases, and outside the peak area, active power flow increases, while reactive power flow is hardly affected.

URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/226400
Collection土木工程与力学学院
Recommended Citation
GB/T 7714
华超锋. 功率流理论及其在杆梁类结构振动分析中的应用[D]. 兰州. 兰州大学,2018.
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