CHAPTER III 

 MECHANICAL VIBRATING SYSTEMS 



3.1. Introduction. — The preceding chapters have been confined to the 

 considerations of simple systems, point sources, homogeneous mediums 

 and simple harmonic motion. Sources of sound such as strings, bars, 

 membranes and plates are particularly liable to vibrate in more than one 

 mode. In addition, there may be higher frequencies which may or may 

 not be harmonics. The vibrations in solid bodies are usually termed as 

 longitudinal, transverse or torsional. In most cases it is possible to con- 

 fine the motion to one of these types of vibrations. For example, the 

 vibrations of a stretched string are usually considered as transverse. It 

 is also possible to excite longitudinal vibrations which will be higher in fre- 

 quency. If the string is of a fairly large diameter torsional vibrations 

 may be excited. The vibrations of a body are also affected by the me- 

 dium in which it is emersed. Usually, in the consideration of a particular 

 example it is necessary to make certain assumptions which will simplify 

 the problem. The mathematical analysis of vibrating bodies is extremely 

 complex and it is beyond the scope of this book to give a detailed analysis 

 of the various systems. The reader is referred to the treatises which have 

 been written on this subject for complete theoretical considerations. It is 

 the purpose of this chapter to describe the most common vibrators in use 

 to-day, to illustrate the form of the vibrations and to indicate the resonant 

 frequencies. 



3.2. Strings. — In all string instruments the transverse and not the 

 longitudinal vibrations are used. In the transverse vibrations all parts 

 of the string vibrate in a plane perpendicular to the line of the string. 

 For the case to be described it is assumed: that the mass per unit length 

 is a constant, that it is perfectly flexible (the stifi^ness being negligible) 

 and that it is connected to massive nonyielding supports, Fig. 3.1. Since 

 the string is fixed at the end, nodes will occur at these points. The fun- 

 damental frequency of the string is given by 



2/ \m 

 36 



