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

) .