47, 48] FORCED VIBRATIONS. 175 



131) A r = - 



X ' s = n 



where 



P, = >V + G>-vO a , tan , = -*. 



n 



Retaining now only the real parts, we have for our solution, 





; 



Thus if the damping coefficients ^ are small, all the oscillations 

 are in nearly the same phase. If the frequency of the impressed 

 force coincides with that of any one of the free oscillations, p i> s = 0, 

 and one factor of the denominator reduces to ^i s) so that if the 

 damping of that oscillation is small, the amplitude is very large, or 

 infinite if there is no damping. This is the case of resonance. 

 (Resonance may also be defined in a slightly different manner as 

 occurring when ip is one of the roots of the equation D(T) = in 

 which all the 's have been put equal to zero. This corresponds 

 with our example in 44. In practical cases the difference is very 

 small.) 



48. Cyclic Motions. Igiioratioii of Coordinates. In certain 

 large classes of motions some of the coordinates do not appear in 

 the expression for the kinetic energy, although their velocities may. 

 For instance in the case of rectangular coordinates, 



the coordinates themselves x, y, z do not appear. In spherical co- 

 ordinates, 41, 133), 



(p does not appear while both r and # do. Further examples are 

 furnished in the case of systems in which throughout the motion 

 the place of one particle is immediately taken by another equal 

 particle moving with the same velocity, as for instance in the case 

 of the system of balls in a ball-bearing (bicycle) or better in the 

 case of a continuous chain passing over pulleys, or through a tube 

 of any form, or by the particles of water circulating through a tube. 



