Page 47 - Servo Motors and Industrial Control Theory -

2.7 Bode Diagram 39

Fig. 2.20 A simple mass–
spring–damper system

C K

M

F
X

d x dx
2
M + C + Kx = F (2.24)
dt 2 dt
It should be noted that the displacement is measured from static equilibrium posi-
tion and, therefore, the weight of mass is ignored.
Taking Laplace Transform from both sides of Eq. (2.22) and assuming zero ini-
tial conditions yields
Ms x + Csx + Kx = F (2.25)
2

With some algebraic manipulation, Eq. (2.23) can be written in standard form as

F
x = K (2.26)
M s + C s +1
2
K K
The natural frequency and damping ratio are given by

ω = K
n
M (2.27)
ζ = C
2 KM