First, the free vibration of the coupled system is discussed. It is mathematically proved that the vibration is always damped.
As an extreme case we have undamped harmonic vibrations. The necessary and sufficient conditions for this case are given. Namely, the coupling factor σ
2=1 and the proper period of the pendulum is equal to that of the galvanometer.
Second, the case, that the galvanometer and pendulum are respectively critically damped, is solved.
Movements of a galvanometer in the early stage of a forced vibration are given.
Third, a realistic example is given, that is, ε
1=
n1, ε
2=
n2,
n1=2π/4.5sec
-1,
n2=2π/7.5sec
-1, σ=0.15, and the circular frequency of a sine motion,
Asinω
t, of the earth, commencing at
t=0, is ω=2π/4sec
-1.
In fig. 1.
φ
0: An incident wave 0.179sin2π/4
t.
φ
1: Free vibration excited by the sudden commencement of the incident wave.
φ
3: Forced vibration caused by the infinite train of the incident waves.
φ: Movements of the galvanometer finally obtained.
View full abstract