﻿and Critical Speeds of Rotors. 141 



The main vibration, represented by cos cot and sin cot, 

 shows a slight change in amplitude, but as before the 

 maximum occurs at the critical speed. 



The term in cos 2cot indicates there is a double fre- 

 quency ripple in the vertical motion y (but not in the 



c 2 e 

 horizontal component x) having an amplitude 2 2 /° 2 ffo? 



that is, ' l . ,.( ., .,) Vn- This rises to a maximum 



at half the first critical speed (when the amplitude changes 

 sign) and again at the first critical speed (when the ampli- 

 tude does not change sign). Noticeable vibration may thus 

 occur at half the critical speed, but it will take place 

 principally in the vertical plane. 



Both components show a triple frequency vibration ex- 

 P'essed by cos Zcot and sin Scot, which reaches a maximum 

 at one-third the critical speed, and the amplitude of the 

 vibration changes sign at that point. This vibration also 

 has a maximum value at the critical speed. 



Points of marked vibration due to lack of proportionality 

 in the deflexion can thus only show themselves when the 

 machine is out-of-balance, and if they become appreciable 

 at all will only occur at half or one-third, etc., of the critical 

 speed. If, however, these fractions of the critical speed 

 correspond to low running speeds, the forces may be so 

 small as not to produce any noticeable effect. 



18. It thus appears that subsidiary critical speeds are only 

 to be expected at half or possibly one-third of the calculated 

 first critical speed, and only then when the subsidiary critical 

 speed is high enough to make the forces appreciable — for 

 example, in the case of a turbo alternator when the speed 

 indicated by the calculation approaches the running speed. 

 All these effects should disappear with perfect balancing, 

 excepting that due to lack of uniformity in the resistance 

 of the shaft or rotor to bending in directions perpendicular 

 to its axis, such as might arise through two-pole slotting of 

 the rotor or through a key-way in the shaft. 



The forces tending to produce vibration are small, and the 

 vibrations arise through a kind of resonance ; as there is a 

 good deal of damping due to air friction and to the move- 

 ment of the shaft in the bearing where the oil exercises a 

 strong damping action, the vibrations may not arise at all. 

 This question is gone into more fully in a later section. 



19. We have now to consider some cases where resonance 

 may arise from causes outside the machine itself, and two 



