The oscillator in musical acoustics
4.3 Lossy oscillators
One could easily introduce a linear loss term into the equation for the lossless oscillator (4.2), but, for collisions of short duration, this will yield results of minor perceptual significance. The real interest in lossy oscillator models in musical acoustics relates to the case of continuous forced excitation, as occurs in bowed string instruments, as well as in woodwind and brass instruments.
4.3.1 The bow
Before examining more realistic bow models, it is useful to look at an archetypical test problem of the form
d2u dt2 = −αφ
du dt
(4.19) Here,α≥0 is a free parameter, and the functionφis a given nonlinear characteristic. Normally, in models of bow friction, the functionφ (η)is antisymmetric aboutη=0, and possesses a region of steep positive slope near the origin (sticking regime), outside of which it is of negative slope (sliding regime). This feature will be explored further in some examples which follow and in Section 7.4, but for reference, some typical examples of such bow characteristics are shown in Figure 4.8. In many models [410], the sticking portion of the curve is in fact of infinite slope—two examples are shown in (a) and (b). A continuous curve, as shown in (c), though less physically justifiable, reasonably approximates this discontinuity, and is somewhat easier to work with numerically.
LOSSY OSCILLATORS 83
−1 −0.5 0
(a) (b) (c)
0.5 h
f(h) f(h) f(h)
h h
1
−1
−0.5 0 0.5 1
−1
−0.5 0 0.5 1
−1
−0.5 0 0.5 1
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
Figure 4.8 Some bow friction characteristics φ (η): (a) a hard characteristic defined by φ (η)=sign(η)e−a|η|, for some a≥0; (b) a hard characteristic with a non-zero limiting sliding friction value, defined byφ (η)=sign(η)
+(1−)e−a|η|
; and (c) a soft characteristic of the formφ (η)=√
2aηe−aη2+1/2.
This equation is of the form of an oscillator without a stiffness term. The variable u itself, undifferentiated, does not appear in the equation, implying a single mechanism for storing energy (through a kinetic term), and that furthermore it will not be possible to obtain bounds onuitself.
Indeed, one could write the above equation in terms of du/dt alone, though it is simpler to retain the second-order form, especially when the above model of a nonlinear excitation is to be coupled to a distributed model. First-order forms equivalent to the above do occur in models of nonlinearities in nonlinear electrical circuit components used in analog synthesizers such as the Moog [165]—see Problem 4.6 for an example. As yet, this remains an unforced problem—the important forcing term will be introduced shortly.
Energy analysis indicates a constraint onφ: multiplying bydu/dt, one obtains, immediately, dH
dt = −αdu dtφ
du dt
with H= 1 2
du dt
2
Clearly then, if the quantityHis to be monotonically decreasing, one must require that the char-acteristicφ (η)satisfy
φ (η)η≥0 or sign(φ (η))=sign(η) (4.20) which is a requirement for passivity. If such a condition does not hold, it becomes possible for the nonlinearity to behave at certain instants as a source of energy, which is certainly not the case in any musical instrument.
As in the case of the lossless oscillator, various difference schemes are possible. Here are two:
δt tu= −αφ (δt−u) (4.21a)
δt tu= −αφ (δt·u) (4.21b)
Scheme (4.21a), the simpler of the two, is clearly explicit, due to the use of a backward difference inside the nonlinear characteristic. As one might guess, however, it is rather difficult to say anything conclusive about its behavior, especially in terms of stability. The scheme (4.21b) permits some such analysis. Multiplying byδt·uyields
δt+h= −α (δt·u) φ (δt·u)≤0 with h= 1
2(δt−u)2 (4.22) Thus, as in the continuous case,his non-negative and monotonically decreasing, and the behavior of scheme (4.21b) is thus stable, as long as the characteristic satisfies (4.20). On the other hand,
−1 −0.5 0
Multiple solutions possible Unique solution always
0.5 1
−1
−0.5 0 0.5 1
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 f(h) 1
y(h)
f(h) y(h)
h h
Figure 4.9 Graphical representation of solutions to the equationφ (q)= −mq+b, for a given bow-like characteristicφ (q)(shown as a solid black line). The linesy(q)= −mq+b, form >0, and for various values ofb, are plotted as dotted black lines. Left, a choice ofmfor which multiple solutions may exist, for certain values of b; and right, a choice ofm for which the solution is always unique, for any choice ofb.
scheme (4.21b) is implicit. Using identity (2.7g), it may be rewritten in the following form:
− 2
αkδt·u+ 2
αkδt−u=φ (δt·u)
Solutions may be examined graphically in terms of the unknownδt·u—see Figure 4.9.
There will always thus be at least one solution to this nonlinear equation, regardless of the form ofφ. Though existence follows immediately, uniqueness does not —depending on the form ofφ, there may be multiple solutions to the equation. The issue of multiple solutions in the case of the bow coupled to the string is an interesting one, and the continuous-time case has been discussed extensively by various authors, beginning with Friedlander [144], and most notably McIntyre and Woodhouse [237]. Here, in the discrete case, the following condition is sufficient:
k≤ − 2
αminηφ(η) when min
η φ(η)≤0 (4.23)
This condition is a purely numerical one, and applies only to scheme (4.21b), though similar conditions exist for other schemes —see Problem 4.7. It is worth keeping in mind that the above conditionhas no bearing on numerical stability, which is already ensured for scheme (4.21b) under condition (4.20). That is, even when multiple solutions do exist, any choice will lead to strictly passive numerical behavior (though such a solution may be physically meaningless). It is, however, of the same form as stability conditions which typically arise in the design of explicit schemes for oscillators. See, e.g., Section 3.2.4 and the following example.
Connection to a mass –spring system and auto-oscillatory behavior
To get a better idea of how the bow actually functions, it is better to move to a more concrete setting involving a coupling of a bow model with a single mass – spring system, as shown in Figure 4.10(a). (In fact, it is only a small further step to connect the bow to a fully distributed string model — see Section 7.4.) Here, the motion of the mass is described by
d2u
dt2 = −ω02u−FBφ (vrel) where vrel=du
dt −vB (4.24)
where FB≥0 is a given control parameter (again, it is the bow force divided by the object mass, and has dimensions of acceleration), and vB is a bow velocity. Notice that it is only the
LOSSY OSCILLATORS 85
0 u
u vB
vrel
0.005 0.01 0.015 0.02 0.025 0.03 0.035 t
t
−2 0.04
−1 0 1 2× 10−3
0
(a) (b)
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
−1
−0.5 0 0.5
Figure 4.10 (a) A bowed mass, as described by (4.24). (b) Simulation results, with f0=ω0/2π=100,FB=4000, andvB=0.2. The friction characteristic is of the form given in Figure 4.8(c), witha=100. Displacement of mass (top), and relative velocityvrel=(du/dt )−vB (bottom), with shaded regions illustrating intervals during which the mass slips from the bow — otherwise it “sticks.”
relative velocityvrel of the bow to the mass which appears in the model. Energy analysis now yields
dH
dt = −FBdu
dtφ (vrel)= −FBvrelφ (vrel)
power dissipated by bow
−FBvBφ (vrel)
power supplied by bow
where H is the Hamiltonian for a linear oscillator, as given in (3.7), and where the terms on the right-hand side of the energy balance may be interpreted as power dissipated and supplied by the bow, as indicated. Because, by (4.20), the dissipated power is negative, one has immediately
dH
dt ≤ −FBvBφ (vrel)≤ |FBvBφ (vrel)|
For many choices of bow characteristic (such as those pictured in Figure 4.8), FB represents a maximum bow force—in other words, the characteristicφis bounded such that
|φ| ≤1 (4.25)
In this case, the energy inequality above may be weakened to dH
dt ≤ |FBvB| −→ H(t )≤H(0)+t|FBvB|
This is a bound on the growth of the solution, purely in terms of the given numbersFBandvB. The above analysis changes very little if the valuesFBandvBare generalized to functionsFB(t )≥0 andvB(t ), representing the gestural control signals of bow force/object mass and velocity.
The obvious generalization of scheme (4.21b) to the case of coupling to a mass – spring sys-tem is
δt tu= −ω20u−FBφ (vrel) where vrel=δt·u−vB (4.26)
The energy analysis of this scheme mirrors that of the continuous-time system above. Using familiar techniques, one arrives at the discrete energy balance
δt+h= −FBvrelφ (vrel)−FBvBφ (vrel)≤ |FBvB|
wherehis the energy of the SHO, as discussed in Section 3.2.6. Again, energy growth is bounded in terms of the input values (or sequences)FBandvB. Two conditions onk, the time step, appear:
k < 2 ω0
, k≤ 2
−max(FB)minηφ(η) when min
η φ(η)≤0 (4.27) The first condition is the familiar stability condition (3.17) for scheme (3.12) for the SHO, which ensures non-negativity of the numerical energy h. The second condition is that required for uniqueness of numerically computed solutions. One could go further and employ a more accu-rate difference staccu-rategy to the linear part of the system, along the lines of the scheme presented in Section 3.3.4, but the principle of stability analysis remains the same. An implementation of scheme (4.26) appears in Section A.3, and makes use of a Newton –Raphson iterative root finder (see Problem 4.8 and Programming Exercise 4.6). For simplicity, the continuous friction charac-teristic shown in Figure 4.8(c) is used —when a discontinuous characcharac-teristic is used, such root finders may also be used, though programming complexity increases somewhat.
The behavior of the bowed mass system is an example of an auto-oscillatory system—given a slowly varying input signal (such as that derived from a player’s gesture), the system can reach a state of continuous oscillation. Even in this rudimentary case, a great variety of complex behavior results, and many of the key features of bowed string dynamics may be observed. The most crucial phenomenon is the stick –slip motion of the mass, as illustrated in Figure 4.10(b). The mass “sticks”
to the bow for an interval as the spring is being extended or compressed, exhibiting little displace-ment relative to the bow, until the force of the spring is sufficient to set the mass into motion, in a direction opposing that of the bow motion, after which it then becomes stuck to the bow once again etc. The resulting displacement waveform is roughly of the form of an asymmetric triangle.
Some other phenomena are illustrated in Figure 4.11. As shown at left, an increase in bow force can lead to a characteristic “sharpening” of the displacement waveform and also an increased period of oscillation —this is a minor effect in bowed string dynamics, and is often referred to as pitch flattening [236]. At right, one may also observe a variation in the length of time necessary to reach steady oscillatory behavior which decreases with increased bow force.
4.3.2 Reed models
Given that the reed mechanism and coupling to an acoustic tube will be seen in great detail in Section 9.3, it is not necessary to present a full treatment in the lumped context —indeed, it may be viewed in terms of a (rather complex!) combination of the various nonlinear mechanisms already discussed in this chapter. At heart it behaves like a valve (see the descriptions in [136] and [78]) or as a lumped, driven, nonlinear oscillator subject to collisions; as in the case of the bow, when coupled to a resonator, auto-oscillations result. Here is a form incorporating many of the features which are by now standard [202]:
d2u dt2 +2σ0
du
dt +ω20u−ωα1
|[u+1]−|α−1
= −F (4.28)
Reading through the terms from left to right, the first three terms, in the absence of the others, describe damped harmonic oscillations, with frequency ω0 and damping parameter σ0. If u(t ) represents the position of the reed, then the oscillations will occur about an equilibrium point at
PROBLEMS 87
0.02 u
u
u
u
u
u 0.04
FB = 200
FB = 500
FB = 1000
FB = 10
FB = 40
FB = 100
0.06 0.08 t t
t
t t
t 0.1
−5 0 5
× 10−4
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
−2 0 2
× 10−4
0.02 0.04 0.06 0.08 0.1
−1 0 1
× 10−3 × 10−4
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
−2 0 2
0.02 0.04 0.06
(a) (b)
0.08 0.1
−2 0 2
× 10−3 × 10−4
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
−4
−2 0 2 4
Figure 4.11 Displacement waveforms for the bowed mass system (4.24) under different choices ofFB, as indicated, and assumed constant. In all cases,vB=0.2,f0=ω0/2π =100 Hz, and the bow friction characteristic is of the form shown in Figure 4.8(c), witha=100. (a) Pitch flattening and waveform sharpening effects with increased bow force, and (b) shortening of the time of onset of a stable auto-oscillatory regime with increased bow force.
u=0. The next term introduces a lossless collision, whenu≤ −1, using a power law characteristic (see Section 4.2.2). Such a collision models, in a rather ad hoc manner, the “beating” of the reed against the mouthpiece lay —notice that it will only become active under rather large oscillations.
Finally, the termF on the right-hand side represents a driving term (i.e., it is related to the control mouth pressure), as well as a loss characteristic—it is similar to that of the bow (see Section 4.3.1), though the characteristic curve is of a very different form. It is, however, dependent on the state of the reed (as well as the tube to which it is coupled) in a rather involved way, and the full description is postponed until Section 9.3.