Sound Modelling
107
characteristic frequency
damping coefficient
quality factor
R
k
m
f
x
Figure 11: Mass-Spring-Damper system
the algebraic relationship
s
2
mX(s) + sRX(s) + kX(s) = F (s) ,
(35)
and we can derive the transfer function between the forcing term f and the
displacement x:
H(s) =
X(s)
F (s)
=
1/m
s
2
+
R
m
s +
k
m
.
(36)
The system oscillates with characteristic frequency
0
=
k/m = 2f
0
and
the damping coefficient is = R/m. The quality factor of the system is Q =
0
/ and it is the number of cycles that the characteristic oscillation takes
to attenuate by a factor 1/e
. The damping coefficient is proportional to
the resonance bandwidth. If we use the bilinear transformation to discretize
the transfer function (36) we obtain the discrete-time system described by the
transfer function
H(z) =
1 + 2z
-1
+ z
-2
mh
2
+ Rh + k + 2(k - mh
2
)z
-1
+ (k + mh
2
- Rh)z
-2
(37)
=
b
0
+ b
1
z
-1
+ b
2
z
-2
1 + a
1
z
-1
+ a
2
z
-2
Therefore, the damped mechanical oscillator can be simulated by means of a
second-order discrete-time filter. For instance, the realization Direct Form I,
depicted in figure 24 of chapter 2, can be used for this purpose. We notice that
there is a delay-free path that connects the input f with the output x, and this
may represent a problem when connecting several simulations of physical blocks
together.
5.4.2
Coupled oscillators
Let us consider the system obtained by coupling the mass-spring-damper oscil-
lator with a second mass-spring system (see figure 12):
m
1
¨
x
1
= -k
1
(x
1
- x
2
) - R( x
1
- x
2
) + f
(38)
m
2
¨
x
2
= -k
1
(x
1
- x
2
) - k
2
x
2
+ R( x
1
- x
2
) .