Modern Control Engineering

Section 3–2 / Mathematical Modeling of Mechanical Systems 65

(a) The force fdue to the dampers is

In terms of the equivalent viscous-friction coefficient beq, force fis given by

Hence

(b) The force fdue to the dampers is

(3–1)

wherezis the displacement of a point between damper b 1 and damper b 2 .(Note that the

same force is transmitted through the shaft.) From Equation (3–1), we have

or

(3–2)

In terms of the equivalent viscous-friction coefficient beq,forcefis given by

By substituting Equation (3–2) into Equation (3–1), we have

Thus,

Hence,

beq=

b 1 b 2

b 1 +b 2

=

1

1

b 1

+

1

b 2

f=beq(y# -x#)=

b 1 b 2

b 1 +b 2

(y# -x#)

=

b 1 b 2

b 1 +b 2

(y# -x#)

f =b 2 (y# -z#)=b 2 cy#-

1

b 1 +b 2

Ab 2 y# +b 1 x#Bd

f=beqAy# -x#B

z# =

1

b 1 +b 2

Ab 2 y# +b 1 x#B

Ab 1 +b 2 Bz# =b 2 y# +b 1 x#

f=b 1 (z# -x#)=b 2 (y# -z#)

beq=b 1 +b 2

f=beq(y# -x#)

f=b 1 (y# -x#)+b 2 (y# -x#)=Ab 1 +b 2 B(y# -x#)

x y

(a)

b 2

x z y

(b)

b 1

b 1
b 2
Figure 3–2
(a) Two dampers
connected in parallel;
(b) two dampers
connected in series.