ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 115 



7. Longitudinal and Torsional Vibrations. — While we are not immediately in- 

 terested in these, for the sake of ready reference the rough formulae are given for 

 corresponding frequencies (steel) : — 



Longitudinal: 71 = 8400// (/ being the length in feet) 

 Torsional: n ^ 5300// (/ being the length in feet) 



These nodes will be characterized by one node in the middle ; the first harmonic will 

 have two nodes and twice the frequency ; the second harmonic will have three nodes 

 and three times the frequency of the fundamental ; etc. 



8. Compounding of Vibrations. — As a general rule a vibration taken at ran- 

 dom is a mixture of the fundamental with several harmonics. Prof. D. C. Miller's 

 delightful book, "Science of Musical Sounds," contains not only a complete bibliog- 

 raphy of this subject, but also many instructive diagrams with clear explanations by 

 which any engineer would profit immensely. Any compound vibration can be de- 

 composed by analysis, or by means of special instruments, into its primary elements 

 (so-called normal modes), fundamental and overtones. Now if there is any disturb- 

 ing periodic force (such as an unbalanced engine), and if its period happens to be 

 the same (or nearly so) as that of one of the normal modes of the free vibration, we 

 have synchronism, where the effect is generally quite out of proportion to the mag- 

 nitude of the disturbing force. For instance, the vibration due to a reciprocating 

 engine is compounded of the fundamental, the first harmonic (of double frequency) 

 and of the second harmonic (of three times the frequency of the fundamental) ; 

 higher harmonics, as a rule, are not felt. Now if any one of these frequencies is the 

 same as any of the natural frequencies of the structure, its effect will be felt in a 

 marked degree. 



The reader, unfamiliar with elastic vibrations in general, is advised to carefully 

 study the fourteenth chapter of Morley's excellent book, "Strength of Materials," 

 where many useful formulae will be found. 



II. morrow's method. 



I. A very interesting method for finding periods of vibrating bars has been pro- 

 posed by Professor Morrow of Bristol, England.* The essential features of this 

 method are as follows. By assuming an equation which completely satisfies the end 

 conditions, we can find both the vibration curve and the period of the fundamental to 

 any required degree of accuracy. Initially, the method gives too small a value of 

 frequency, which improves with further steps. When the density and flexural 

 rigidity of the bar are variable from point to point in its length, the finding of 

 periods by ordinary methods of analysis is often a problem of utmost difficulty; while 

 the method proposed by Professor Morrow leads to much easier solution. The main 

 principle lies in the fact that if a bar is vibrating so that every point in it has the 

 same period (in other words, in one of its normal modes) then the ratio of the ac- 

 celeration to the displacement is constant for all points. 



*PhU. Mag.. 1905, pp. 113-125. 



