弦、杆、环的横振动理论与实验研究  

Theoretical and experimental study of transverse vibration of string,beam and ring

作　　者：方奕忠 崔新图 沈韩 黄臻成 冯饶慧 廖德驹 庞晓宁 FANG Yi-zhong;CUI Xin-tu;SHEN Han;HUANG Zhen-cheng;FENG Rao-hui;LIAO De-ju;PANG Xiao-ning(National Demonstration Center for Experimental Physics Education,School of Physics,SunYat-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》2023年第6期32-39,共8页College Physics

基　　金：国家自然科学基金(61871410);中山大学质量工程项目(中山大学教务[2022]20号,74130-12220011)资助。

摘　　要：本文对弦的横振动方程进行了严格推导,得出波速的解析解,并在共振时测量了波长和共振频率的实验值,理论与实验相吻合.本文对杆的横振动方程亦进行了严格推导,对一端卡住、另一端自由时进行了数值求解,得出共振时容许的前几个本征频率的表达式及相应共振波节位置,给出通解,实验中测出了不同长度时不锈钢薄细长条的基频及前几阶泛音共振频率,测出的波节位置与理论值一致,并求出不锈钢长条的杨氏模量,与标准值相符.进一步推广到圆形钢丝环的小振动情形,实验上测出了圆环共振时的频率及波长,求出截面为圆形的细钢丝的杨氏模量,与现有值基本符合.From the strict derivation of transverse vibration of string,the explicit solution of wave velocity is obtained.Through measuring the wavelength and eigen frequency of resonance,the experiment value of wave velocity is also obtained,which is well agreed with the theoretical value.The transverse vibration equation of beam is also solved exactly.The simulation of the problem which one end is clamped and the other is free,is performed.The expressions of first several resonant eigen frequencies and the sites of standing-wave nodes are obtained so as to get the general solution.Further,the fundamental frequencies and resonant frequencies of the previous order overtone of steel slender strips under different lengths are measured.The co-ordinates of standing-wave nodes are observed and equivalent to the theoretical results,and the Young’s modulus of the strips are calculated,which are well corresponded with standard values.Furthermore,the case of small radial vibration of ring is popularized.The resonant frequencies and wavelengths are also measured,and the elastic modulus of circular cross section ring are obtained,where are also accorded with the existing values.

关 键 词：弦的横振动 杆的横振动 圆环的径向振动 本征频率 杨氏模量 

分 类 号：O329[理学—一般力学与力学基础]