tion —acos nt. Then the forced vibrations are determined by the equation 
dz , da 
—__—= — 7,” & —a — acos nt*). 
dt? ‘dt 
Its solution is 
a 
w——cos(nt + Pp), ee © «+ CY) 
q ) 
if 
n> —n, = Geosp, ng =qsing 
From this one finds 
(he 
and 
1 1 
= —— : eee 
q V (n? ny rug? 
If therefore there is a cause, which produces an acceleration 
alternating with the frequency 7, the amplitude of this acceleration 
has to be multiplied by 1,48 in order to obtain the amplitude of 
the forced vibration *). 
If now the acceleration is represented by the curve of fig. 4, so 
that the area of each of the peaks is a measure for the velocity S 
arising from one impulse, then on development in a Fourier series 
5 48 F 8 
the amplitude « of the first term is equal to a and that of the forced 
vibrations becomes 
4S 
1,48 . ZB IO iS bev, ay. AT « = wild) 
la al 
We shall use this to determine the multiplication by resonance. 
If the cylinder when at rest suddenly receives a velocity S at the 
time 0, the first deviation is (here we may neglect the damping) 
T,. 1,912 
ek) 
2 250 
The effect is therefore multiplied somewhat more than 10 times. 
In my experiments the period of the external force was smaller 
than that of the free vibration. As to the phase of the impressed 
vibrations, it might be supposed to be what it would have been 
without damping, or if this exists, at a large distance from the equality 
5 J dz . ' 
1) In this equation x, 7 etc. may be regarded as angle of rotation, angular 
at 
velocity etc. but also as deflexion, velocity etc. on the scale. 
2) If we wholly neglect the damping, we find 
Thus the damping has an influence of a few percent on the amplitude-factor, 
