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NUMERICAL SIMULATION OF SPRING REVERBERATION

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Virtual analog modeling of spring reverberation presents a chal-lenging problem to the algorithm designer, regardless of the par-ticular strategy employed. The difficulties lie in the behaviour of the helical spring, which, due to its inherent curvature, shows characteristics of both coherent and dispersive wave propagation. Though it is possible to emulate such effects in an efficient manner using audio signal processing constructs such as delay lines (for coherent wave propagation) and chains of allpass filters (for dis-persive wave propagation), another approach is to make use of di-rect numerical simulation techniques, such as the finite difference time domain method (FDTD) in order to solve the equations of motion directly. Such an approach, though more computationally intensive, allows a closer link with the underlying model system— and yet, there are severe numerical difficulties associated with such designs, and in particular anomalous numerical dispersion, requir-ing some care at the design stage. In this paper, a complete model of helical spring vibration is presented; dispersion analysis from an audio perspective allows for model simplification. A detailed description of novel FDTD designs follows, with special attention is paid to issues such as numerical stability, loss modeling, numer-ical boundary conditions, and computational complexity. Simula-tion results are presented.
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Proc. of the 16th Int. Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, September 2-4, 2013
NUMERICAL SIMULATION OF SPRING REVERBERATION
Stefan Bilbao,
Acoustics and Audio Group
University of Edinburgh
Edinburgh, UK
sbilbao@staffmail.ed.ac.uk
ABSTRACT
Virtual analog modeling of spring reverberation presents a chal-
lenging problem to the algorithm designer, regardless of the par-
ticular strategy employed. The difficulties lie in the behaviour
of the helical spring, which, due to its inherent curvature, shows
characteristics of both coherent and dispersive wave propagation.
Though it is possible to emulate such effects in an efficient manner
using audio signal processing constructs such as delay lines (for
coherent wave propagation) and chains of allpass filters (for dis-
persive wave propagation), another approach is to make use ofdi-
rect numerical simulation techniques, such as the finite difference
time domain method (FDTD) in order to solve the equations of
motion directly. Such an approach, though more computationally
intensive, allows a closer link with the underlying model system—
and yet, there are severe numerical difficulties associated with such
designs, and in particular anomalous numerical dispersion, requir-
ing some care at the design stage. In this paper, a complete model
of helical spring vibration is presented; dispersion analysis from
an audio perspective allows for model simplification. A detailed
description of novel FDTD designs follows, with special attention
is paid to issues such as numerical stability, loss modeling, numer-
ical boundary conditions, and computational complexity. Simula-
tion results are presented.
1. INTRODUCTION
Virtual analog modeling, for electronic effects and synthesis mod-
ules has seen an enormous amount of work in recent years [1].
Somewhat less investigated has been the case of electromechanical
effects, especially when the mechanical components involved have
a distributed character (i.e., they cannot be modelled as lumped).
Prime examples are plate reverberation, and the system of interest
in this paper, spring reverberation [2, 3, 4].
Part of the difficulty in adequately simulating spring reverber-
ation lies in the complexity of the model of a helical spring which,
even in the linear case, possesses features which are very much
unlike those of what might seem to be similar systems in mu-
sical acoustics, such as the string, or ideal bar. See [5, 6, 7, 8]
for presentations of the dynamical system governing spring vibra-
tion. Helical structures, due to their inherent curvature possess
the characteristics of both coherent wave propagation, giving rise
to discrete echoes, and dispersive wave propagation, giving the
response a diffuse character as well. Nevertheless, modeling of
spring reverberation has proceeded apace, with simulation meth-
ods based on allpass networks [9, 10, 11], and, for simplified mod-
This work was supported by the European Research Council, under
grant StG-2011-279068-NESS.
els, through standard time stepping procedures such as finite dif-
ference schemes [12].
A model of spring vibration is presented in Section 2, accom-
panied by dispersion analysis, illustrating regions of coherent and
dispersive wave propagation. Finite difference schemes are intro-
duced in Section 3, and simulation results and performance analy-
sis are presented in Section 4.
2. HELICAL SPRING MODELS
A helical structure of the type found in typical spring reverberation
units, is characterized by its geometrical parameters as illustrated
at left in Figure 1, and namely: R, the coil radius in m, r, the
wire diameter in m, α, the pitch angle, and L, the unwound spring
length in m. Also necessary are parameters describing the mate-
rial, namely E, Young’s modulus in Pa, ν, Poisson’s ratio, and ρ
the material density in kg/m3. A general model of the linear dy-
namics of such helical structures is due to Wittrick [5, 6] and is
most easily written in terms of time tand a curved coordinate s
along the axis of the spring (see Figure 1 at right) as a system of
12 variables:
sξ=cos2(α)
RJξ+Eθ+1
GAγ Kp (1a)
sθ=cos2(α)
RJθ+1
EI Lm (1b)
sm=cos2(α)
RJm +Ep +ρIMttθ(1c)
sp=cos2(α)
RJp +ρA∂ttξ(1d)
Here, the vector variables ξ,θ,mand p,
ξ=
u
v
w
θ=
θu
θv
θw
m=
mu
mv
mw
p=
pu
pv
pw
(2)
represent, respectively, displacements, rotation angles, moments
and forces relative to orthogonal local coordinates (γu, γv, γw)at
location salong the spring axis. A=πr2is the cross sectional
area of the spring, and sand trepresent partial differentiation
with respect to the coordinates sand t, respectively (and tt =
2
t). The system matrices Jand Eare given by
J=0 tan(α)1
tan(α) 0 0
1 0 0 E=0 1 0
100
0 0 0(3)
DAFX-1
Proc. of the 16th Int. Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, September 2-4, 2013
Figure 1: A helical spring. Left, side view illustrating the coil radius R, wire radius rand pitch angle α. Right, view illustrating the
orthogonal coordinates [γu, γv, γw]at curvilinear coordinate salong the spring wire axis.
and K,Land Mare diagonal matrices given by
K="1 0 0
0 1 0
0 0
E#L="1 0 0
0 1 0
0 0 EI
GIφ#M="1 0 0
0 1 0
0 0 Iφ
I#(4)
where G=E/2(1 + ν)is the shear modulus, γis a shear area
correction, and where Iand Iφare transverse and polar moments
of inertia, respectively. For a spring of circular cross section, γ=
0.88,I=πr4/4, and Iφ= 2I=πr4/2.
2.1. Scaled Form
In nondimensional form, i.e., introducing
s=s
s0
t=t
t0
ξ=ξ
s0
θ=θp=s2
0p
EI m=s0m
EI
(5)
where s0=R/ cos2(α)and t0=pρA/EI R2/cos4(α)yields
the scaled system (after removing primes):
sξ=Jξ+Eθ+EI
GAγs2
0
Kp (6a)
sθ=Jθ+Lm (6b)
sm=Jm +Ep +I
As2
0
Mttθ(6c)
sp=Jp +ttξ(6d)
This model is quite refined, and can be classified as thick—indeed
it reduces to the thick model of beam vibration, due to Timoshenko
[13], in the limit of vanishing curvature.
2.2. Reduction to a Thin Model
The model described in the previous sections is a complete de-
scription of the linear vibration of a helical spring; for audio rate
simulation of spring reverberation, however, it is unneccesarily
complex. To this end, note that the factors I/As2
0in (6a) and
(6c) are proportional to r2/R2, and are thus extremely small for
springs typically found in reverberation units. Neglecting these
terms leads to the form:
sξ=Jξ+Eθsθ=Jθ+Lm (7a)
sm=Jm +Ep sp=Jp +ttξ(7b)
This system in 12 variables (i.e., in 4 three-vector variables)
can be reduced to a system in eight variables, as:
At˜
ζ=Q˜
p˜
p=Q˜
mDt˜
m=Q˜
φ˜
φ=Q˜
ζ
(8)
where the reduced variables ˜
ζ,˜
φ,˜
mand ˜
pare defined as
˜
ζ=tv
w˜
φ=tθu
θv˜
m=mv
mw˜
p=pu
pv(9)
and where the matrix differential operators Q,Q,A, and Dare
defined as
Q=sµ∂s
µ1 + ssQ=µ ∂s
1ss µ∂s(10)
A=1 0
0 1 ssD=1 0
0bss(11)
Note that the displacement umay be recovered from the system as
u=sw. The scaled system as a whole now depends only on the
three parameters µ,band λ, defined as
µ= tan(α)b=EI
GIφ
λ=L/s0(12)
This system is similar to that presented in [8]; as a prelude to
numerical implementation, note that by elimination of ˜
φand ˜
pthe
system can be rewritten in terms of ˜
ζand ˜
mas
At˜
ζ=sR˜
mDt˜
m=sR˜
ζsR=QQ(13)
where Rmay be written explicitly as
R=2µ1µ2+ss
1µ2+ss 2µ(1 + ss)(14)
Finally, by elimination of ˜
m, a system in ˜
ξ= [v w]Talone
may be written as
Att˜
ξ=ssRD1R˜
ξ(15)
where here, D1is to be interpreted as the inverse of the differen-
tial operator D.
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Proc. of the 16th Int. Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, September 2-4, 2013
2.3. Dispersion Relations
As a justification for the use of the simplified model (8), as op-
posed to the full model (1), it is useful to examine the two sys-
tems assuming wave like solutions—which is appropriate, as both
systems are constant coefficient linear systems. To this end, solu-
tions with harmonic time/space dependence ej(ωt+β s)for all com-
ponents in both cases, where ωis temporal frequency, and βis
wavenumber (both nondimensional). Partial differential operators
tand sthus behave as multiplicative factors, i.e.
t=jω ∂s=jβ (16)
Under this assumption, system (1) reduces to a 12 ×12 sys-
tem of equations in ωand β, and thus possesses 12 dispersion
relations of the form ω(β); these are generally grouped in pairs
(corresponding to leftward and rightward propagation), so there
are essentially six independent such relations.
Consider now a spring of dimensions typical to spring rever-
beration units. As shown in Figure 2, the six dispersion relations
for the full system span a large range of frequencies; there are only
two, however, which lie comfortably within the audio range, with
the other four in the range above 100 kHz. The reduced model
(8) very accurately models the two relations in the audio range,
as shown in Figure 2. It is safe to use the reduced model under
virtually any spring configuration to be found in a reverberation
unit.
101102
10−4
10−2
100
102
f(kHz)
β(1/m)
Figure 2: Log-log plot of dispersion relations f(β), in Hz, in di-
mensional form, for a steel spring of coil radius of R= 5 mm, a
wire radius of r= 1 mm, with a pitch angle of α= 5. Black:
the six dispersion relations corresponding to the full model (1),
with the upper two indistinguishable in this plot. Red: the two dis-
persion relations for the reduced system (8). The limit of human
hearing at f=20 kHz is indicated by a blue line.
The two principal dispersion relations of interest possess sev-
eral general features. See Figure 3 at left, showing such rela-
tions for a typical spring under variation in the pitch angle α.
Both curves exhibit a main “hump"; the curves track each other
closely, but one of the two falls to zero at dimensionless wavenum-
ber β=p1 + µ2; this wavenumber corresponds to a wavelength
of exactly one full turn of the spring, and at this wavenumber, rigid
body rotation (i.e., at frequency ω= 0) is permitted. The second
curve shows increasing deviation from the first in this region as the
pitch angle αis increased. Above this critical wavenumber, both
curves approach those corresponding to dispersive wave propaga-
tion in an ideal thin bar.
More useful, from the perceptual standpoint, is the examina-
tion of group velocity curves vg=dω/dβ (see the right panel of
Figure 3); flat regions of the curves correspond to coherent wave
propagation, and thus to spectral regions over which echoes will
be perceived. For more on the interpretation of dispersion rela-
tions for the helical spring system, see [14].
0 50 100 150 200 250
0
2
4
6
8
10
β(1/m)
f(kHz)
0
4
8
0 50 100 150 200 250
−1000
−500
0
500
1000
1500
2000
β(1/m)
vg(m/s)
0
4
8
Figure 3: Left: Principal dispersion relations (dimensional) for the
helical spring of parameters given in the caption to Figure 2, under
variation in the pitch angle α, as indicated. Right: Associated
group velocities vg.
2.4. Energy, Boundary Conditions, Forcing and Output
The total energy stored in the spring is a useful quantity in numer-
ical design—see, e.g., [15] for more on this topic. In the present
case of a lossless system, it is conserved, provided boundary con-
ditions are chosen as lossless as well.
An energy balance for the system (8) may be written as
d
dt H=Bs=λ− Bs=0 (17)
where the total energy His defined as
H=1
2Zλ
0
ζ2
v+ζ2
w+(sζw)2+m2
v+bm2
w+(smw)2ds (18)
where the first three terms under the integral correspond to kinetic
energy density in the three coordinate directions, and the latter
three to potential energy. The boundary term consists of six terms:
B=ζvpv+ (sζw)
| {z }
ζu
pu+ζw(µpvspu+stζw)
| {z }
pw
(19)
+mvφv+ (smw)
| {z }
mu
φu+mw(µφvsφu+b∂stmw)
| {z }
φw
When evaluated at the endpoints of the domain, as in (17), Bhas
the interpretation of power supplied at the boundaries. Under un-
forced conditions, if B= 0 at the boundaries, then the system
is lossless—i.e., dH/dt = 0. Natural boundary conditions are
that one constituent of each of the six products in the boundary
term above vanish, generalizing frequently occurring conditions
such as free, clamped or simply supported to the case of a heli-
cal structure. There are obviously many combinations (4096, in
fact, considering that there are 64 possible choices of such natural
boundary conditions at each end of the spring)!
The terms in (ζ , p)above correspond to power supplied through
forcing along the coordinate directions, and those in (m, φ)to
power supplied through twisting action about the three coordi-
nate directions. Under forced conditions in a spring reverberation
unit, though a combination of linear and rotational forcing is cer-
tainly present, the assumption here will be that the forcing occurs
through the former mechanism, so that one may specify pu,pvor
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Proc. of the 16th Int. Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, September 2-4, 2013
pw(through the implicit expression in (19)) as an input signal, and
leave the corresponding velocity component unspecified.
A simple means of taking output is by directly reading values
of the spring velocity ˜
ζat the location s=λ(or, if working with
the simplified system (15) in displacements, their time derivatives,
which will scale roughly with observed output signals).
2.5. Losses
The question of losses within the spring, due to viscothermal ef-
fects is an important one for reverberation purposes; however, it
has not seen any investigation, experimental or otherwise, to the
knowledge of this author. A basic three parameter loss model, al-
lowing for uncoupled damping effects is as follows:
Att˜
ξ=ssRD1R˜
ξσv0
0σwσusst˜
ξ(20)
where σu, σv, σw0are loss parameters corresponding to vibra-
tion in the γu,γvand γwcoordinates respectively (and taking into
account the relationship between uand win the reduced form).
3. FINITE DIFFERENCE TIME DOMAIN METHODS
There are several different equivalent systems presented in Section
2; the full model in eight variables (8), a system in four variables
(13) and finally a system in two displacements alone (15) (as well
as the lossy system (20)). The first is useful in the construction
of simulation methods, and in properly modeling boundary condi-
tions; the latter are the forms which are most useful in a practical
implementation.
3.1. Grid Functions
In moving to a discrete time formulation, the first step is the choice
of sample rate Fs, and the resulting time step k= 1/Fs.
First consider system (15), in the displacements ˜
ξalone. A
grid spacing hmay be chosen, such that λ/h =N, for integer
N, and subject to stability conditions (see Section 3.4). The grid
function ˜
ξn
lthen represents an approximation to ˜
ξ(s, t)at times
t=nk, for integer n, and at locations s=lh, for l= 0,...,N.
See Figure 4, showing the lattice of grid point corresponding to ˜
ξn
l
in black.
The system (13) in velocities ˜
ζand moments ˜
mpermits an
interleaved discretization, similar to that which occurs in FDTD
schemes for electromagnetics [16, 17]. One may define the grid
functions
˜
ζn+1
2
l˜
ζ|t=(n+1
2)k,s=lh ˜
mn
l+1
2
˜
m|t=nk,s=(l+1
2)h
(21)
as illustrated by white points in Figure 4. ˜
ζn+1
2
lis defined for
l= 0,...,N, and ˜
mn
l+1
2for l= 0,...,N 1.
Finally, for the complete system (8), grid functions correspond-
ing to ˜
φand ˜
pare necessary. It is appropriate, given the structure
of system (8), to split the components of these vector variables
onto distinct grids as:
˜
φn+1
2
u,l+1/2˜
φn+1
2
v,l ˜
pn
u,l ˜
pn
v,l+1
2(22)
˜
ξ, ˜pu
˜
ξ, ˜pu
˜
ξ, ˜pu
˜
ξ, ˜pu
˜
ξ, ˜pu
˜
ξ, ˜pu
˜
ξ, ˜pu
˜
ξ, ˜pu
˜
ξ, ˜pu
˜m, ˜pv
˜m, ˜pv
˜m, ˜pv
˜m, ˜pv
˜m, ˜pv
˜m, ˜pv
˜m, ˜pv
˜m, ˜pv
˜m, ˜pv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
ζ, ˜
φv
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
˜
φu
x= 0hx=h
2x=hx=3h
2x= 2hx=5h
2
t= (n1)k
t= (n
1
2)k
t=nk
t= (n+1
2)k
t= (n+ 1)k
t= (n+3
2)k
t= (n
3
2)k
Figure 4: Computational grids for FDTD schemes for the helical
spring. Black points indicate a regular lattice for displacements
˜
ξn
land white points indicate an interleaved lattice in velocities
˜
ζn+1
2
land moments ˜
mn
l+1
2. A scheme for the full system requires
an additional lattice of points, indicated in green.
where ˜
φn+1
2
v,l and ˜
pn
u,l are defined for l= 0,...,N, and ˜
φn+1
2
u,l+1/2
and ˜
pn
v,l+1
2for l= 0 ...,N 1. The lattices of grid points cor-
responding to these grid functions are as indicated in Figure 4.
3.2. Difference Operators
For a given grid function fn
l(where here, land nare either integer
or half-integer), forward and backward approximations δt+and
δtto the time derivative tare defined as
δt+fn
l=1
kfn+1
lfn
lδtfn
l=1
kfn
lfn1
l(23)
Similarly, forward and backward approximations δs+and δsto
the spatial derivative sare defined as
δs+fn
l=1
h(fn
l+1 fn
l)δsfn
l=1
h(fn
lfn
l1)(24)
Approximations to higher derivatives may be obtained through
composition of the above operations. For example, centred ap-
proximations to the second time and space derivatives tt and
ss may be obtained as the operator products δtt =δt+δtand
δss =δs+δs.
The main spatial differential operators for the spring system
are as given in (10), and may be directly transferred to spatial dif-
ference operators as
Qd,±=δs±µδs±
µ1 + δssQ
d,±=µ δs±
1δss µδs±
Ad=1 0
0 1 δssDd=1 0
0bδss(25)
Note that due to the interleaved character of the grid functions, the
differential operators Qand Qhave been replaced by forward
and backward difference operations Qd,±, and Q
d,±respectively.
If the reduced systems (13) or (15) are to be used, then the dif-
ference operator Rd, corresponding to Rmay be defined in terms
of either of these pairs of operators as
δs±Rd=Q
d,±Qd,±(26)
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Proc. of the 16th Int. Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, September 2-4, 2013
and takes the explicit form
Rd=2µ1µ2+δss
1µ2+δss 2µ(1 + δss)(27)
3.3. Interleaved Scheme
Assembling the various difference approximations to the operators
in (8) leads immediately to the following interleaved form:
Adδt˜
ζ=Q
d,˜
p˜
p=Qd,˜
m(28)
Ddδt+˜
m=Q
d,+˜
φ˜
φ=Qd,+˜
ζ
Reduced forms corresponding to (13) and (15) can be written
as
Adδt˜
ζ=δsRd˜
mDdδt+˜
m=s+Rd˜
ζ(29)
and
Adδtt˜
ξ=δssRdD1
dRd˜
ξ(30)
where as before, D1
dis to be interpreted as the inverse of the
difference operator Dd(i.e., in implementation it will become a
matrix inversion or linear system solution).
Notice in particular that the single variable scheme in (30) de-
pends only on centered difference operators and is thus second
order accurate. All of the above schemes are necessarily implicit,
due to the action of the operators Adand Dd(in implementation,
these lead to sparse linear system solutions to be carried out at each
time step).
3.4. Stability and Numerical Boundary Conditions
The determination of explicit numerical stability conditions for
the scheme above is considerably more difficult in the case of the
present scheme (or indeed any scheme for this system) than for
other related systems (such as, e.g., the ideal string or ideal bar
[15]). The difficulty is fundamental: for a given time step k, one
would like to find a minimum value hmin of the grid spacing such
that the scheme does not exhibit explosive growth.
One approach to finding a numerical stability condition is the
familiar frequency domain approach due to von Neumann [18].
In this case all grid functions are assumed to exhibit harmonic
time/space dependence of the form ej(lhβ+nkω ), for angular fre-
quency ωand wavenumber β, and for half integer land n. Con-
sidering, for simplicity, the centred one-variable form (30), the op-
erators δtt and δss transform according to
δtt =⇒ −4 sin2(ωk/2)
k2δss =⇒ −4 sin2(βh/2)
h2(31)
and thus the scheme (30) possesses the characteristic equation
sin2(ωk/2)I2=k2
h2sin2(βh/2) ˆ
A1
dˆ
Rdˆ
D1
dˆ
Rd(32)
where I2is the 2×2identity matrix, and where ˆ
Ad,ˆ
Ddand ˆ
Rd
are the wavenumber-dependent matrices obtained through replace-
ment of δss by the factor 4 sin2(βh/2)
h2in the definitions (25) and
(27). As such, the necessary stability condition follows as
eig k2
h2sin2(βh/2) ˆ
A1
dˆ
Rdˆ
D1
dˆ
Rd1(33)
for all βwith 0βπ/h. Ideally, one would like to be able to
determine a closed form expression for the minimal grid spacing
hmin(k), such that the above condition is satisfied for all h
hmin(k)—in audio applications, for best results, for a given time
step k, it is best to choose has close to hmin as possible [15].
Unfortunately, such a closed form condition is not available in the
present case, and thus hmin must be determined numerically.
In the limit of high sample rates (or small time steps), the min-
imal grid spacing approaches the following bound:
lim
k0hmin(k) = 2k(34)
The square root dependence is typical of schemes for stiffsystems
such as the ideal bar (indeed, at high frequencies, the helical spring
system approaches that of the ideal bar). See, e.g., [15].
Energy conservation methods [19] are a more powerful means
of obtaining stability conditions, as proper numerical boundary
conditions may also be determined (though the difficulty in ob-
taining a closed form expression for hmin remains). The idea is
to obtain an energy balance analogous to (17), such that numerical
energy is conserved from one time step to the next—numerical sta-
bility then amounts to finding conditions under which the numeri-
cal energy is positive semi-definite in the grid functions, allowing
solution growth to be bounded.
Though there is not space here to give a full treatment of en-
ergy methods, a discrete energy balance can be shown to be:
δt+Hd=Bd,0− Bd,N (35)
where one expression for Hn
d, the discrete conserved energy at
time step n, is
Hn
d=h
2N
X
l=0
ζn
v,l2+ζn
w,l2+
N1
X
l=0
1
h2ζn
w,l+1 ζn
w,l2
+
N1
X
l=0
mn+1
2
v,l+1
2
mn1
2
v,l+1
2
+bmn+1
2
w,l+1
2
mn1
2
w,l+1
2
+1
h2
N
X
l=0
mn+1
2
w,l+1
2mn+1
2
w,l1
2mn1
2
w,l+1
2mn1
2
w,l1
2
where the primed sum indicates a factor of 1/2 applied at l= 0
and l=N. Such an expression mirrors that in (18), though in
this case, the terms in ˜
mare not necessarily positive. The deter-
mination of conditions under which the above expression is non-
negative as a whole (by relating ˜
mback to ˜
ζthrough (29)) is dif-
ficult, and amounts to the same problem discussed with reference
to frequency domain methods above.
The boundary terms Bd,0and −Bd,N again indicate power
supplied at the endpoints of the domain, at l= 0 and l=N; the
expressions are rather lengthy, and will not be included here—the
general form, however, is the same as that for the continuous sys-
tem from (19), and is made up of six products of grid functions
evaluated at the boundary. The important point is that one may
ensure numerical losslessness, under unforced conditions, by re-
quiring that one member of each of the six pairs vanishes. See
Section 4.2. It is important to point out that, in contrast to the con-
tinuous case, the expression for the numerical energy Hdis not
unique—each choice leads to a distinct set of numerical boundary
conditions.
3.5. Numerical Dispersion
The scheme (28) (or the equivalent reduced forms) for the heli-
cal spring exhibits numerical dispersion—i.e., propagation speeds
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deviate from those of the model system. Such behaviour may be
analyzed through the characteristic equation (32), which possesses
two solutions ω(β). See Figure 5, showing numerical dispersion
relations at a variety of audio sample rates. The dispersion curves
converge (with increasing sample rate) rather slowly to those of the
model system. An appeal to more delicate modeling is in order—
see Section A for some more involved numerical designs targeting
this deficiency.
Figure 5: Numerical dispersion relations, for a variety of audio
sample rates, for scheme (28) for the helical spring system, for a
steel spring of coil radius R= 5 mm, wire radius of r= 1 mm,
and pitch angle α= 5.
4. SIMULATION RESULTS AND PERFORMANCE
ANALYSIS
In this section, some simulation results are presented, for springs
of dimensions typically found in reverberation units.
4.1. Response to an Impulsive Excitation
Figure 6 shows snapshots of the time evolution of the state of a he-
lical spring, when subjected to a short impulsive excitation aligned
nearly with the coil axis (at right), and transverse to it (at left)—in
this case, the values of the forcing function ˜
pin the appropriate
direction specified at the boundary at l= 0. Note in particular
the conversion of thevibrational energy almost instantaneously to
vibration in the other coordinate directions.
4.2. Energy Conservation
The scheme (28), as discussed in Section 3.4, conserves energy to
machine accuracy under lossless conditions, as long as appropriate
conservative boundary conditions are applied. See Figure 7, illus-
trating the variation in numerical energy for a spring subjected to
an impulsive excitation—note that energy is quantized to multi-
ples of machine precision. Such a property, beyond being useful
in stability analysis, serves as an excellent debugging tool—any
errors in the implementation will almost certainly lead to anoma-
lous variation in the numerical energy. When losses are present,
numerical energy, at least under the model will be monotonically
decreasing.
4.3. Spring Responses
In this section, spectrograms of simulated spring output responses
are shown. In all cases here, the spring is steel, of length L= 5
m, with R= 4 mm, r= 0.2mm, and with a pitch angle of 1.7;
Figure 6: Snapshots of the time evolution of a helical spring, sub-
ject to an impulsive excitation, at times as indicated. The spring
is steel, with R= 4 mm, and r= 0.2mm and a pitch angle of
α= 5. Left: excitation along coordinate direction γv. Right:
excitation along coordinate direction γw. Displacements are ex-
aggerated, for visibility.
0 50 100 150 200 250 300 350 400
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10−15
time step n
Normalized energy variation
Figure 7: Numerical energy variation for the spring of parameters
as given in the caption to Figure 6, using scheme (28) at a sample
rate of 44.1 kHz. Multiples of machine epsilon are shown as grey
lines.
output is drawn from velocity ζwin the γwcoordinate direction.
Other parameters will be varied here.
The most important consideration, given the discussion of nu-
merical dispersion in Section 3.5, is that of anomalous wave speed,
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particularly for echoes. Figure 8 shows spectrograms for spring
output under successively larger sample rates. In general, the spec-
trograms exhibit a complex region, under a cutoff frequency (the
frequency at the top of the hump in the plot of the dispersion rela-
tion in Figure 3), over whichmultiple families of dispersive echoes
are visible. Above the cutoff, a single family of dispersive chirp-
like echoes persists. It is this family which is most affected by
numerical dispersion, as should be evident in the plots in Figure
5, and at 44.1 kHz, wavespeed is anomalously slow. Thus, unless
a scheme with reduced dispersive properties can be devised, such
schemes will produce anomalous resultsat low audio rates (though
they may be used at oversampled rates). See Section A for some
possible variants on the scheme presented in Section 3.3.
Figure 8: Spectrograms of output responses, under different
choices of the sample rate, as indicated.
Loss, though discussed only in passing in this article, obvi-
ously plays a major role in determining the characteristic sound of
a spring reverberation unit. Considering the simple lossy model in
(20), it is simple to arrive at a generalization of scheme (30), by re-
placing instances of tby a centered approximation (δt++δt)/2,
and ss by δss. It is also possible to show that such a perturbation
leads to monotonic energy loss.
Consider two cases, as illustrated in Figure 9, selecting for loss
along one of the two coordinate directions γvor γw; as is easily
visible, loss along the transverse coordinate γvhas a much greater
effect in terms of energy decay.
Figure 9: Spectrograms of output responses, under different
choices of loss coefficients σvand σw, as indicated. The sample
rate is 88.2 kHz.
4.4. Computational Cost
Computational cost for the scheme given in Section 3.3 is high,
but not extreme; clearly it depends strongly on the sample rate, so
there is some motivation to look forschemes with reduced dispersion—
see Section A. As an example, consider a spring of dimensions
typical to a reverberation device. In Table 1, simulation times,
for one second output are given, for a single core Matlab imple-
mentation on a standard laptop computer. Notice in particular that
the grid size scales roughly with Fs, and computation time with
F3/2
s—reflecting the limiting stability condition from (34).
Table1: Number of grid points Nand calculation time, in seconds,
for one second output at various sample rates, for a spring with
R= 4 mm, r= 0.2mm, L= 5 m and α= 1.7.
FsNcalculation time (s)
44 100 1257 18.13
88 200 1639 50.90
176 000 2205 141.0
5. CONCLUDING REMARKS
A model for a helical spring has been presented here; the continu-
ous model (8) presented, while extremely accurate in the case of a
lossless, unforced spring of dimensions typical in spring reverber-
ation units, is still lacking in several features which are necessary
if one is to arrive at a complete model of a spring reverberation
device.
Loss has not been considered here in any depth; as has been
mentioned, models of loss in helical structures are not readily avail-
able. Even an empirical study of losses (by measuring, say, band-
withdths of spectral peaks, as in the case of a string of vibrating
bar) is complicated, as there are two overlapping families of modal
frequencies (corresponding to the two distinct dispersion relations
for the system); furthermore, each mode is itself a mixture of lon-
gitudinal and transverse vibration, so a direct association with loss
characteristics of simpler non-curved systems such as strings and
bars is not immediately forthcoming.
The forcing mechanism also has not been described here. In
a typical spring reverberation unit, the excitation is a permanent
magnet attached to one end of the spring, and driven by an electro-
magnetic field (the strength of which is proportional to the input
signal). The exact details of the forcing, and how it couples to
the various components of displacement and rotation is as yet un-
known, and is in need of experimental investigation. The analysis
of boundary conditions for the spring system given in Section 2.4,
however, does at least provide a framework for arriving at a com-
patible forcing term, which will certainly have the character of a
mass-spring system.
The numerical method presented here iswell-behaved numer-
ically under all possible boundary terminations, which may be eas-
ily associated with those of the underlying model system; energy
conservation, in the lossless case, has been used as a design prin-
ciple. In terms of accuracy, however, other problems emerge—
all are due to the appearance of a new length scale in the case
of a tightly curved structure such as a spring found in a typical
reverberation unit. The dispersion characteristics possess impor-
tant features at relatively high wavenumbers (corresponding to the
length of one turn of the helix); in uncoiled structures, such high
wavenumbers are associated with high temporal frequencies, and
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thus there is less perceptual importance associated with them. In
this case, however, the temporal frequencies of interest lie directly
in the low audio range, and thus great care is necessary in numer-
ical design over the whole range of wavenumbers. One simple
remedy is to operate at oversampled rates; but certain approaches
based on parameterized FDTD methods (see Section A) offer some
degree of control over such perceptually important features.
Given that the input and output locations for the spring rever-
beration system is generally always taken from the same locations
(i.e., the ends of the spring), and also that the system is linear and
time invariant, an interesting alternative to FDTD methods might
involve modal approaches, as in, e.g., the standard helical spring
literature [6], and also as used in sound synthesis [20, 21, 22]. The
obvious advantages relative toFDTD methods are (a) reduced run-
time cost, (b) the possibility of obtaining exact behaviour, and (c)
better control over loss. There are disadvantages as well; one must
precompute the modal shapes and frequencies offline, which is po-
tentially a very large undertaking, as well as store them, for each
new set of parameters dand µ, and for a particular set of boundary
conditions; still, though, this would seem to be one of the most
suitable systems for modal-based effects modeling.
A. IMPROVED FDTD DESIGNS
Numerical dispersion is a strong effect for the scheme presented
in Section 3.3, which is the simplest possible choice for the helical
spring system. Other choices are possible, including free parame-
ters allowing for tuning of the algorithm. To this end, consider the
following modifications of the spatial difference operators given in
(25) and (27):
Rd=2µ1µ2+ (1 + ǫ1h2)δss
1µ2+ (1 + ǫ1h2)δss 2µ(1 + (1 + ǫ1h2)δss)
Ad=1 0
0 1 (1 + ǫ2h2)δssDd=1 0
0d(1 + ǫ3h2)δss
Three parameters ǫ1,ǫ2and ǫ3have been introduced—note that all
three are multiplied by factors of h2, so that the resulting scheme
(29), when the above operators are employed, remains consistent
with the helical spring system [18].
Under appropriate choices of the free parameters (perhaps cho-
sen through an optimization procedure), much better behaviour
can be obtained at audio sample rates. See Figure 10. Interest-
ingly, such schemes can also be cheaper computationally than the
unparameterized scheme, as hmin is generally larger (leading to a
smaller grid size).
B. REFERENCES
[1] V. Välimäki, U. Zölzer, and J. O. Smith, Eds., 2010, IEEE Trans-
actions on Audio Speech and Language Processing: Special Issue on
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[2] L. Hammond, “Electrical musical instrument,” Feb. 2 1941, US
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[3] A.C. Young and P. It, “Artificial reverberation unit,” Oct. 8 1963, US
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[4] J.D. Stack, “Sound reverberating device,” Mar. 9 1948, US Patent
2,437,445.
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national Journal ofMechanical Sciences, vol. 8, pp. 25–47, 1966.
0 50 100 150 200 250 300
0
5
10
15
20
25
β(1/m)
f(kHz)
model
simple scheme
parameterized scheme
Figure 10: Numerical dispersion relations, at 44.1 kHz, for a
spring with R= 5 mm and r= 1 mm, and with pitch an-
gle α= 5: for the simple scheme (in red) and for a param-
eterized scheme (in green), for ǫ1= 0.084,ǫ2=0.291 and
ǫ3=0.364.
[6] J. Lee and D. Thompson, “Dynamic stiffness formulation, free vi-
bration and wave motion of helical springs, Journal of Sound and
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[8] L. Della Pietra and S. della Valle, “On the dynamic behaviour of
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[10] J. Parker, “Efficient dispersion generation structures for spring reverb
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[11] J. Parker V. Välimäki and J. Abel, “Parametric spring reverberation
effect,Joural of the Audio Egineering Society, vol. 58, no. 7/8, pp.
547–562, 2010.
[12] S. Bilbao and J. Parker, “A virtual model of spring reverberation,
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[13] K. Graff, Wave Motion in Elastic Solids, Dover, New York, New
York, 1975.
[14] S. Bilbao and J. Parker, “Perceptual and numerical issues in spring
reverberation models,” in Proceedings of the International Sympo-
sium on Musical Acoustics, Sydney, Australia, 2010.
[15] S. Bilbao, Numerical Sound Synthesis, John Wiley and Sons, Chich-
ester, UK, 2009.
[16] K. Yee, “Numerical solution of initial boundary value problems in-
volving Maxwell’s equations in isotropic media,” IEEE Transactions
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[17] A. Taflove, Computational Electrodynamics, Artech House, Boston,
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[18] J. Strikwerda, Finite Difference Schemes and Partial Differential
Equations, Wadsworth and Brooks/Cole Advanced Books and Soft-
ware, Pacific Grove, California, 1989.
[19] B. Gustaffson, H.-O. Kreiss, and J. Oliger, Time Dependent Problems
and Difference Methods, John Wiley and Sons, New York, New York,
1995.
[20] D. Morrison and J.-M. Adrien, “Mosaic: A framework for modal
synthesis,” Computer Music Journal, vol. 17, no. 1, pp. 45–56, 1993.
[21] J.-M. Adrien, “The missing link: Modal synthesis,” in Representa-
tions of Musical Signals, G. DePoli, A. Picialli, and C. Roads, Eds.,
pp. 269–297. MIT Press, Cambridge, Massachusetts, 1991.
[22] L. Trautmann and R. Rabenstein, Digital Sound Synthesis by Physi-
cal Modeling Usingthe Functional Transformation Method, Kluwer
Academic/Plenum Publishers, New York, New York, 2003.
DAFX-8
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Digital sound synthesis has long been approached using standard digital filtering techniques. Newer synthesis strategies, however, make use of physical descriptions of musical instruments, and allow for much more realistic and complex sound production and thereby synthesis becomes a problem of simulation. This book has a special focus on time domain finite difference methods presented within an audio framework. It covers time series and difference operators, and basic tools for the construction and analysis of finite difference schemes, including frequency-domain and energy-based methods, with special attention paid to problems inherent to sound synthesis. Various basic lumped systems and excitation mechanisms are covered, followed by a look at the 1D wave equation, linear bar and string vibration, acoustic tube modelling, and linear membrane and plate vibration. Various advanced topics, such as the nonlinear vibration of strings and plates, are given an elaborate treatment. Key features: Includes a historical overview of digital sound synthesis techniques, highlighting the links between the various physical modelling methodologies. A pedagogical presentation containing over 150 problems and programming exercises, and numerous figures and diagrams, and code fragments in the MATLAB® programming language helps the reader with limited experience of numerical methods reach an understanding of this subject. Offers a complete treatment of all of the major families of musical instruments, including certain audio effects. Numerical Sound Synthesis is suitable for audio and software engineers, and researchers in digital audio, sound synthesis and more general musical acoustics. Graduate students in electrical engineering, mechanical engineering or computer science, working on the more technical side of digital audio and sound synthesis, will also find this book of interest.
Article
A set of 12 partial differential equations pertaining to helical springs is solved for free vibrations by the transfer matrix method. The dynamic transfer matrix including the axial and the shear deformations and the rotational inertia effects for any number of coils is numerically determined up to any desired precision in an efficient way. It is proved that the coefficients of the characteristic determinant of the dynamic differential matrix, [D], with odd-numbered subscripts are equal to zero which is based on the peculiarity that the traces of the same matrix with odd powers are all equal to zero. This important property of [D] has been the essence of the developed solution algorithm. The validity of the computer program coded in Fortran-77 has been verified by means of comparisons with the results given in literature. Next, the effects of the helix angle, the boundary conditions, the number of coils, and the ratio of (cylinder diameter/wire diameter) on the free vibration frequencies are investigated.