Introduction to Acoustics

570 Part E Music, Speech, Electroacoustics
Part E 15.2
To investigate such effects experimentally, Galluzzo
and Woodhouse [15.54, 65] have recently developed
a dynamically controlled bowing machine with active
feedback, providing programmable control of both
downward bow pressure and bow speed. This enables
reliable and reproducible results to be made over a very
wide range of possible playing parameters, extending
Guettler’s original measurements.
Viscoelastic Friction
Recent measurements have shown that the frictional
model assumed in these investigations is over-simplistic.
The force between the bow hairs and the string is maintained
by a thin layer of rosin which coats them both.
Rosin is a rather soft, sticky substance, with a glassto-liquid
transition not far above room temperature,
resulting in viscoelastic properties, which are very sensitive
to temperature (Smith, Woodhouse [15.66]). As
the bow slides past the bow hair, the frictional forces
will heat the rosin and hence reduce its viscoelasticity
frictional properties. During the sticking regime,
with no work being done at the bow–string interface,
the rosin will cool down and the friction will increase.
The frictional forces are therefore hysteretic and will
be strongly dependent on past history within a given
period of string vibration. Woodhouse et al. [15.68]
and Smith [15.69] have investigated this hysteretic behaviour
in some detail using rosin-coated glass rods.
The hysteretic properties shown in Fig. 15.36 were deduced
from measurements at the two supported ends
of the string. Woodhouse [15.70] subsequently extended
Friction coefficient
1
0.8
0.6
0.4
0.2
0
–0.15 –0.1 –0.05 0 0.05
Velocity (m/s)
Fig. 15.36 Measured hysteretic frictional force between
string and a glass bow coated with rosin, with the dashed
line indicating previously assumed velocity dependence
(after Smith and Woodhouse [15.66])
his computational models to incorporate the hysteretic
frictional properties. Somewhat surprisingly, this more
realistic model made little qualitative difference to the
predicted behaviour. Such measurements contribute to
our understanding of the physical processes underlying
viscoelastic properties of various coatings and lubricants
and have become an important tool in the field
of tribology (studies of friction).
15.2.4 Bridge and Soundpost
We now consider the role of the bridge and soundpost
in providing the coupling between the vibrating strings
and the vibrational modes of the body of instruments
of the violin family. We also consider the influence of
such coupling on the modes of string vibration, which
involves a discussion of the very important influence of
damping on the normal modes of any coupled system.
Bridges
Many plucked and struck stringed instruments, such as
the piano or guitar, use a rather low solid bridge to support
the strings and transfer energy directly from the
transverse string vibrations perpendicular to the supporting
soundboard or front-plate of the instrument. The
bridge needs only to be sufficiently high to prevent the
strings from vibrating against the fingerboard or shell
of the instrument. This is also true for the Chinese twostring
violin, the erhu, which is held and played so that
the bow excites string vibrations perpendicular rather
than parallel to the stretched snake-skin membrane supporting
the bridge and strings. The strings of a harp are
attached to an angled sounding board, so that transverse
string vibrations in the plane of the strings couple directly
to the perpendicular vibrations of the supporting
soundbox Fletcher and Rossing ([15.5], Sect. 11.2).
For such instruments, the bridge and other string terminations
play a relatively insignificant acoustic role,
3060 6100 985 2100 Hz
Fig. 15.37 The lowest in-plane resonant modes and frequencies
of violin and cello bridges (after Reinicke [15.67]).
The arrows represent the vibrational directions of the
bowed outer and middle strings