相互垂直的简谐振动合成及李萨如图形的渐进变化  

COMPOSITION OF PERPENDICULAR SIMPLE HARMONIC MOTIONS AND THE ASYMPTOTIC VARYING OF LISSAJOURS-FIGURE

作　　者：栾其斌 王丰[2] LUAN Qibin;WANG Feng(Faculty of Electronic Information and Electrical Engineering,Dalian University and Technology,Dalian,Liaoning 116024;School of Physics,Dalian University and Technology,Dalian,Liaoning 116024)

机构地区：[1]大连理工大学电子信息与电气工程学部,辽宁大连116024 [2]大连理工大学物理学院,辽宁大连116024

出　　处：《物理与工程》2021年第4期37-40,共4页Physics and Engineering

基　　金：国家自然科学基金(11505021,11975068);国家重点研发计划(2017YFE030052);中央高校基本科研业务费(DUT17RC(4)34)。

摘　　要：本文通过引入不同的时间尺度,对二维空间中两个垂直方向的简谐振动合成做了定量和定性的分析。虽然当两个振动的频率比为无理数时,在严格意义上不存在完全稳定的李萨如图形,但如果振动的频率比接近某一李萨如图形对应的频率比时,通过时间尺度分析,合成运动可以分解为一系列不同相位的李萨如图形及其之间的渐进变化。此外,本文利用振动的相位和两个垂直方向的振动方程的特点,将二维空间中的运动延拓到三维空间,指出这种李萨如图形的渐进变化为三维空间下的匀速旋转,且旋转速度取决于振动频率比与其临近有理数的接近程度。By introducing different time scales,this paper makes a quantitative and qualitative analysis of the composition of simple harmonic vibrations in two vertical directions in two-dimensional space.When the frequency ratio of two perpendicular simple harmonic motion is irrational number,there is no completely stable Lissajous figure in strict sense.However,when the irrational frequency ratio close to a rational number,the synthetic motion can be decomposed into a series of Lissajous figures with different phases and their gradual changes through time scale analysis.In addition,according to the characteristic of phase and the oscillation equation,the motion in two-dimensional space is extended to three-dimensional space.It is pointed out that the gradual change of Lissajous figure is uniform rotation in three-dimensional space and the rotation speed depends on how close the vibration frequency ratio is to its adjacent rational number.

关 键 词：简谐振动 李萨如图形 时间尺度 渐进变化 

分 类 号：O321[理学—一般力学与力学基础]