Active feedforward control of flexural waves in
an Acoustic Black Hole terminated beam
J. Cheer, K. Hook and S. Daley
Institute of Sound and Vibration Research, University of Southampton,
University Road, Southampton, SO17 1BJ, United Kingdom
E-mail: j.cheer@soton.ac.uk
Abstract. Acoustic Black Holes (ABHs) are structural features that are
typically realised by introducing a tapering thickness profile into a structure
that results in local regions of wave-speed reduction and a corresponding
enhancement in the structural damping. In the ideal theoretical case, where
the ABH tapers to zero thickness, the wave-speed reaches zero and the wave
entering the ABH can be perfectly absorbed. In practical realisations, however,
the thickness of the ABH taper and thus the wave-speed remain finite. In
this case, to obtain high levels of structural damping, the ABH is typically
combined with a passive damping material, such as a viscoelastic layer. This
paper investigates the potential performance enhancements that can be achieved
by replacing the complementary passive damping material with an Active
Vibration Control (AVC) system in a beam-based ABH, thus creating an Active
ABH (AABH). The proposed smart structure thus consists of a piezo-electric
patch actuator, which is integrated into the ABH taper in place of the passive
damping, and a wave-based, feedforward AVC strategy, which aims to minimise
the broadband flexural wave reflection coefficient. To evaluate the relative
performance of the proposed AABH, an identical AVC strategy is also applied
to a beam with a constant thickness termination. It is demonstrated through
experimental implementation, that the AABH is able to achieve equivalent
broadband performance to the constant thickness beam-based AVC system, but
with a lower computational requirement and a lower control effort, thus offering
significant practical benefits.
Keywords: Acoustic Black Hole, Active Control, Reflection Coefficient Submitted
to: Smart Mater. Struct.
Active Acoustic Black Hole terminated beam 2
1. Introduction
The ‘Acoustic Black Hole’ (ABH) effect is a mechanism for attenuating flexural
vibrations in structures, such as beams [1, 2, 3, 4, 5, 6, 7, 8, 9] or plates
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and inherently provides a lightweight
vibration control solution. Although the term ‘Acoustic Black Hole’ has a number
of definitions in different fields, this paper focuses on the structural design feature
that aims to control flexural structural waves. The ABH effect is achieved by
introducing a smooth impedance change into a structure and this introduces a
local reduction in the flexural wave speed. This has typically been realised using
a power law taper [1], which governs the thickness of the beam within the ABH
section and can be described as
h(x) = ε
l
taper
− x
l
taper
µ
+ h
tip
, (1)
where x is the coordinate position, l
taper
is the length of the taper, µ is the power
law and h
tip
is the tip height. Figure 1 shows a diagram of an example power
law taper terminating a beam. Theoretically, if the taper were to decrease to
an infinitely small tip height, then the wave speed would reduce to zero and
there would be no reflection from the beam termination [21]. In practice, an
infinitely small tip height is unachievable and so the wave speed does not reach
zero. However, the reduced wave speed results in a reduced flexural wavelength
and it has been shown that it is thus possible to achieve a very low level of reflection
from the beam termination by adding a thin layer of passive damping material to
the taper [1, 12, 17, 22].
ABH$Junction ABH$Tip
!
"#$%&
'
(
")$
(*+,
Figure 1. A diagram of an ABH termination on the end of a beam. h(x) is the
height function, l
taper
is the taper length, b is the taper width and h
tip
is the
tip height.
Due to the lightweight nature of ABHs and the high levels of vibration
control that they can achieve, there have been many studies into their design
Active Acoustic Black Hole terminated beam 3
and optimisation, which have been reviewed in [23]. It has been shown through
both simulation and experimental studies that ABHs can be tuned by changing the
geometrical properties of the taper, such as the tip height, taper gradient and taper
length [19, 24, 25, 26, 27, 28, 29]. It has also been shown that these parameters
are interdependent and they can thus be optimised when considering different
practical limits due to manufacturing or the intended application [6, 30, 31]. An
alternative method of tuning the behaviour of an ABH was proposed in [18], in
which a thermally controlled damping layer was utilised. Although this method
enabled the properties of the ABH to be effectively tuned, it required the system
to be located in a thermal chamber to allow the temperature to be adjusted and,
therefore, further work is required to enable utilisation in practical applications.
In general, ABHs have a low frequency cut-on limit, below which their
performance has been shown to significantly degrade [14]. The frequency above
which the ABH becomes effective can be linked to the frequency at which the
flexural wavelength becomes comparable to the taper length [10]. Although it is
possible to increase the taper length to lower the cut-on frequency, this is not
always achievable in practical applications where space is limited. Attempts to
overcome the low-frequency limit using alternative designs have been proposed,
such as using a spiral ABH [3, 32]. However, such designs are more complex to
manufacture and integrate into existing structures. Alternatively, it has been
shown that the low frequency performance of an ABH can be enhanced by
exploiting geometrical nonlinearities, which serve to induce energy transfer from
low to high frequencies [33]. This behaviour, however, is inherently amplitude
dependent and relies on the introduction of higher harmonic content, both of
which may not be suitable for all applications.
An alternative and highly flexible approach to extending the low frequency
performance of an ABH is to integrate Active Vibration Control (AVC) technology
into the ABH and an initial simulation-based investigation into this smart
structures concept has been presented in [34] and some details have been described
in [35]. AVC is an effective solution for the control of structural vibration when
there are restrictions on the size and weight of the control treatment [36], and thus
presents a complimentary solution to the passive ABH. The current paper presents
a full experimental investigation into the realisation of an Active ABH (AABH) for
the termination of a beam and provides new insight into the potential advantages
that are obtained by this smart structure over a purely active control system
realisation by combining the passive and active control system components. In
order to control the reflection from the end of the beam, a feedforward wave-based
Active Acoustic Black Hole terminated beam 4
control strategy is adopted, as previously developed for the realisation of an active
anechoic termination in a constant thickness beam [37, 38, 39, 40]. Following a
detailed review of the wave-based control strategy in Section 2, Section 3 describes
the realisation of the AABH terminated beam, as well as a standard constant
thickness beam with an active termination. The design and performance of the
AABH and active beam are then also compared in Section 3. This experimental
investigation compares the performance of the two strategies, as well as their
requirements in terms of both electrical power and the computational demand,
which governs the required Digital Signal Processing (DSP) capacity. Section 4
presents a real-time experimental assessment of the performance of the AABH in
comparison to the active beam and conclusions are drawn in Section 5.
2. Wave-Based Active Control
AVC is an effective and versatile solution for the control of structural disturbances
when there are restrictions on the size and weight of the control treatment. AVC
uses a control force to generate additional vibration that destructively interferes
with the primary disturbance. If time-advanced information about the disturbance
is known, then the AVC system can be implemented using a feedforward control
architecture and this can be realised as a digital controller that can be adapted
in real-time to reduce the primary disturbance [41]. In the beam-based ABH
literature, the performance of an ABH is generally assessed in terms of its reflection
coefficient and an ideal ABH performs as an anechoic termination, absorbing
the incident energy so that there is no reflection. Similarly, AVC systems can
be used to absorb incident or reflected waves propagating along a beam and,
as introduced above, a wave-based feedforward AVC control strategy has been
proposed to generate an anechoic beam termination [37, 38, 39, 40]. Wave-based
control can provide broadband attenuation and requires no prior knowledge about
the modal behaviour of the system [40]. As a result, when compared with a global
control strategy the number of error sensors required to control high order modes
is lower and wave-based control can, therefore, reduce both the space and weight
required by the AVC system. In order to investigate the potential advantages of
integrating an active solution into the design of an ABH, this section will describe
a wave-based feedforward AVC strategy. Previous versions of this control strategy
have been described in [37, 38, 39, 40], but were only applied to a constant thickness
beam profile. Section 2.1 first describes the real-time wave decomposition process
and then Section 2.2 describes the controller formulation.
Active Acoustic Black Hole terminated beam 5
2.1. Wave Decomposition
To perform wave-based active control, the primary disturbance must first be
decomposed into the incident and reflected wave components. This can be achieved
by expressing the flexural acceleration at a point on the beam at a single frequency
in terms of the incident and reflected, or positive and negative propagating and
near-field wave components as
A(x, t) = −ω
2
(φ
+
e
−ikx
+ φ
+
N
e
−kx
+ φ
−
e
ikx
+ φ
−
N
e
kx
)e
iωt
, (2)
where ω is the angular frequency, k is the flexural wavenumber, x is the position
along the beam relative to a user-defined origin, φ
+
is the incident propagating
wave, φ
+
N
is the incident near-field wave, φ
−
is the reflected propagating wave and
φ
−
N
is the reflected near-field wave [36, 40]. In the following, the time dependence
term in Equation (2), e
iωt
, will be suppressed for clarity. In order to extract the
individual wave components from the acceleration response of the beam described
by Equation (2), it is necessary to utilise an array of sensors, with the number
of sensors required being equal to the number of wave components [36]. Thus,
according to Equation (2), it would be necessary to employ an array of four
structural sensors to extract the four wave components. However, by placing the
sensor array sufficiently far from structural excitations or impedance changes, such
as the tapering thickness profile in the ABH terminated beam, the near-field terms
in Equation (2) will be small in magnitude compared to the far-field terms, such
that they can be neglected [37, 38]; this assumption and the resulting limitations
are discussed further in Section 2.1.1. Assuming that the near-field components
can be neglected, the two propagating wave components can be decomposed using
two sensors, which in this case are realised as accelerometers. Figure 2 shows the
two accelerometers located on a beam, which are used to decompose the measured
response into the positive and negative propagating wave components. By referring
to Equation (2) and neglecting the near-field components, the response measured
at each accelerometer can be expressed at each frequency as
"
A
1
(ω)
A
2
(ω)
#
= −ω
2
"
e
ik∆/2
e
−ik∆/2
e
−ik∆/2
e
ik∆/2
#"
φ
+
(ω)
φ
−
(ω)
#
, (3)
where A
1
and A
2
are the complex amplitudes of the acceleration measured at each
of the two sensors, and ∆ is the accelerometer spacing shown in Figure 2. The
2 × 2 matrix of exponential terms in Equation (3) can be inverted so that φ
+
and
φ
−
can be expressed as a function of the accelerations measured at the two points.
