重力作用和共振激励下水平杆竖向振动的研究  

Investigation on vertical vibration of horizontal rod under gravitation and resonant exciting

作　　者：方奕忠 崔新图 沈韩 黄臻成 冯饶慧 廖德驹 庞晓宁 李梦钰 FANG Yi-zhong;CUI Xin-tu;SHEN Han;HUANG Zhen-cheng;FENG Rao-hui;LIAO De-ju;PANG Xiao-ning;LI Meng-yu(National Demonstration Center for Experimental Physics Education,School of Physics,Sun Yat-sen University,Guangzhou,Guangdong 510275,China)

机构地区：[1]中山大学物理学院物理学国家级实验教学示范中心,广东广州510275

出　　处：《大学物理》2025年第1期27-33,共7页College Physics

基　　金：国家自然科学基金(61871410);广东省本科高校教育质量与教学改革工程建设项目(粤教高函(2024)30号);中山大学质量工程项目(中山大学教务〔2023〕96号)资助.

摘　　要：本文用本征函数展开法和分离变量法严格求解了水平放置的均匀杆在重力作用下一端保持水平且作垂直小振幅简谐振动,另一端自由时的竖向振动的解析解,并求出了杆作稳态简谐振动时所有的固有频率(容许频率).进一步得到当振动端垂直振动的频率趋向于杆的固有频率时,随着时间的增加,杆将出现波节和波腹的共振现象,且共振时的频率及波节和波腹位置分布与一端固支、一端自由时杆的固有频率及波节和波腹位置分布一致,与重力的作用无关.这表明当共振发生时,杆的振动端由于振幅够小,可以看作是波节,与实验相符.本文最后用拉普拉斯变换对求解过程中的初值问题进行了验证.By using the method of eigen function expansion and separation of variables,under gravitational action,the definite problem of vertical vibration of horizontal uniform rod which one end maintain level but harmonically oscillate perpendicularly and the other end is free is solved strictly,all the natural frequencies of stable harmonic oscillation of level rod are acquired.It is also found that along with the time increasing continuously,when the frequency of vibration which act on the left end approach to the value of natural frequency of level rod,the rod will resonate which the nodes and wave loops are appeared.The resonant frequency and the place distribution of nodes and antinodes distinguishedly agree to the results of level rod while the left end is clamped and the right end is cantilevered,which is independent to the gravity action.It is indicated when the resonance happens,the place of left end of rod where the vibration source at can be regarded as wave node,which is acquiesced by the existing experiment.At last,the fruit of initial value problem in the process of solving is verified by Laplace transformation method.

关 键 词：水平杆的竖向振动 本征函数展开法 分离变量法 固有频率 共振现象 

分 类 号：O326[理学—一般力学与力学基础] O411[理学—力学]