1903.] on Vibration Problems in Emjineering Science. 237 



2. The moments of these forces about any point in the axis of 

 rotation shall have no resultant. 



The model is running now in a state in which neither condition 

 is satisfied, and the consequent vibration is manifest. 



Adding this mass I know, by previous calculation, that the two 

 conditions are separately satisfied. Eunning it now it seems to stand 

 quite still, although it is supported on these flexible springs. 



There are few people who have not had some experience of steam - 

 ship vibration. In recent years the vibration problem has forced 

 itself into notice because of the increase in engine speeds and the 

 relative lightness of the hulls. The generally-held belief, not so 

 many years ago, was that the vibration in ships was produced entirely 

 by the propeller. To disprove this current notion Mr. Yarrow carried 

 out a series of beautiful and costly experiments on a first-class tor- 

 pedo-boat. The propeller was removed and the boat moored in still 

 water, and the engines were run under different conditions of 

 balancing. The consequent vibration disclosed itself by the ripples 

 surrounding the hull. Unbalanced engines caused violent disturb- 

 ance ; balanced, the ripples were smoothed out in the way you can 

 see from the photographs of the two cases on the screen, the slide 

 for which has been kindly lent by Mr. Yarrow. These experiments 

 proved conclusively that one cause of steamship vibration may be 

 removed by balancing the engines. 



Hitherto I have tacitly assumed that the pistons move with simple 

 harmonic motion. Really the forces required for the acceleration of 

 the pistons are different from those in simple harmonic motion because 

 of the obliquity of the connecting rod. Whatever law the force may 

 follow, however, it has one property, namely, that it is continuous 

 and periodic, and may thus be represented by a Fourier series. It 

 may be shown that the force on any one piston is given by an expres- 

 sion of the form — 



F = Mw;V { cos (9 + A cos 2^ + B cos 4(9 + C cos 6^} 



The values of the co-efficients have been worked out by Mr. 

 Macalpine, and for the case where the rod is 3 J times the length 

 of the crank, the series becomes — 



F = Mw?V I cos ^ -h 0-29 cos 2^ - 0-006 cos 4t6 + 0-0002 cos 6^ } 



Looking at these terms, their value decreases very rapidly after the 

 second. By the methods I have indicated the first term of the series 

 is eliminated from all the series belonging to the respective pistons. 

 This is called balancing for primary effects, and is conditioned by 

 four simultaneous equations. 



To eliminate the second term four more equations have to be 

 satisfied, making eight in all. These equations are now on the 

 screen, and the next slide shows the solution. 



This solution, as you will readily perceive, is not one from which 

 a practicable four-crank engine can be made. If the last two of the 



