PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Synthesis of the inerter by direct acceleration feedback
Neven ALUJEVIĆ1; Ivan ĆATIPOVIĆ1; Hinko WOLF1; Nikola VLADIMIR1
1

University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Croatia

ABSTRACT
Physical realisations of the inerter are often such that they must be either large scale, i.e. rack and pinion
inerters, or they inherently include additional elements in parallel or in series with the inerter, i.e. shunted
electromechanical transducers or active force feedback inerter realisations, or alternatively they bring alonga large parasitic damping, i.e. fluid-based inerters. In this study inerter is realised by feeding back thesubtracted outputs of two accelerometers attached to a reactive force actuator terminals. Although in theory
such feedback loop is unconditionally stable due to the collocated sensor-actuator arrangement, in practice it
may not exhibit good stability properties due to the lack of duality between the sensors and the actuator in
conjunction with the internal dynamics of the transducers. Therefore in this paper dynamics of seismic
accelerometer sensors and an electrodynamic actuator are fully incorporated into a theoretical model of suchan inerter. The inerter model is coupled to a two degree of freedom mechanical system in order to study thestability of the feedback loop and the maximum possible synthesisable inertance. The results indicate that it
is crucial to have a highly damped accelerometer resonance in order to achieve good stability and large
synthesised inertance.
Keywords: Interter, Active vibration control, Vibration isolation

1. INTRODUCTION
Inerter is a one-port, two-terminal element in mechanical networks which resists relative
acceleration across its two terminals. The coefficient of resistance, the inertance, is measured in
kilograms. The inerter fills an empty niche enabling a complete analogy between mechanical andelectrical networks, the electrical analogue of the inerter being the capacitor (1,2). In the frameworkof mechanical network analysis, it is typically assumed that the inerter behaves in an idealised way,i.e. that it can be represented through its inertance only. Such idealisations are ordinarily assumed forelements like springs, dampers, inductances or resistors in lumped parameter mechanical or electrical
networks. However, a realistic element, for example a helical spring, can itself exhibit a rich dynamic
behaviour (3). This is equally true for other elements of lumped parameter mechanical systems.
Fidelity of single-parameter representations of physical mechanical network elements depends on how
the element is designed in practice.
Mechanical inerter designs include rack and pinion inerters (4), ball-screw inerters (5), and helicalfluid channel inerters (6). One of the most important characteristics of any physical realisation of theinerter is the ratio of its inertance to its mass. This ratio is normally required to be large so as toenhance inertia effects of lightweight structures without significantly increasing their mass. In
mechanical inerter designs the inertance can be several hundred times larger than the mass of theinerter itself (4). However, such inerters are typically mid to large scale and are not suitable forvibration control purposes in small scale applications due to their large dimensions and large stroke.
Furthermore, effects such as friction, stick-slip of the gear pairs, or the elasticity of the gears andconnecting rods are inevitably present in gear-train inerter constructions. On the other hand, a
relatively large parasitic non-linear damping characterises the fluid-based inerters (6).
Another class of inerter realisations are the electromechanical inerters. In these systems
electomechanical transducers are shunted with appropriate electrical impedance s at their electrical
ports in order to generate inertance-like effects at their mechanical ports. Small scale
1neven.alujevic@fsb.hr

77

electromechanical transducers are characterised by a relatively low energy conversion efficiency (7),so it is necessary to use non-Foster shunt circuits in order to compensate for losses in the transducers(8). This makes the approach active, which on one hand requires energy and on the other a careful
regard of the stability and robustness of the system. Nevertheless, it is possible to synthesise an ideal
inerter element connected to additional elements that occur as a side-effect of using a particularshunting technique. For example, by shunting a voice coil transducer with a certain negative
impedance electrical circuit, an inerter connected in series with a parallel spring damper-pair can besynthesised (8). The additional lumped parameter elements in the equivalent mechanical network may
or may not be desirable.
Further active approaches to realise the inerter include the force feedback approach (9,10). Here apair of collocated reactive actuators and a force sensor are used so as to feed back the output of theforce sensor through both single and double integrators to drive the actuator. In such a way the inerter
can be synthetized which is connected in series with a damper (10).
In this paper, an attempt is made to synthesise the inerter by direct accele ration feedback control
system. Outputs of two accelerometers mounted at the two terminals of a force actuator are subtracted
to form a relative acceleration error signal which is amplified through an adjustable gain and fed tothe actuator. In this way it may be possible to realise a small-scale inerter with an actively tuneable
inertance without additional elements in parallel or in series to the inerter. Classical inertial
accelerometers and a small scale voice-coil actuator are assumed. Although the control approach is
physically well-founded, its application is not straightforward. For example, the relative acceleration
sensor is collocated to the reactive force actuator, but the two transducers are not dual (11,12), i.e.
they are not complementary in terms of mechanical power. As a result, the inherent frequency responseof the sensor-actuator transducers can inhibit the stability of the feedback loop, as discussed inreference (13), for example.
A fully coupled electromechanical model of a two degree of freedom mechanical system equippedwith the described active control loop is formulated in this paper. The dynamics of the inertial
accelerometers and the electrodynamic actuator are modelled in detail. The coupled model is used to
study the stability of the feedback loop and to assess the range of synthesizable inertances that could
be used to isolate simple harmonic discrete vibrations coming from a flexible base to sensitiveequipment mounted on it. In the second section of the paper the model problem is described and themathematical model is formulated. In the third section the stability and the response of the active
system are discussed, which is followed by the conclusions section.

2. MODEL PROBLEM
The mechanical system considered is a lumped parameter two degree of freedom (DOF) system
shown in Figure 1. This system is a representation of the vibration isolation problem in which thesensitive equipment, m2, is suspended with a spring k2 and a damper c2 onto a flexible basecharacterized by the mas m1, stiffness k1 and a damper c1. Dynamic excitation is applied to the basemass through the simple harmonic primary force fp. The formulation presented in this paper considers
time-harmonic functions, which are defined in complex form f t^

Re f Z eiZ t` , where f Z

is the1 . As normally done incomplex amplitude of the function Z is the circular frequency and ivibration studies, the formulation that follows thus refers to the complex amplitudes f Z of thetime-harmonic functions f t and, for brevity, the frequency dependence is omitted.
As discussed in, for example, (14), with the inclusion of the inerter of inertance b2 into thesuspension, Figure 1 (b), a zero can be assigned to the transfer admittance between the primary force,
k2 b2 . If the damping
fp , and the displacement of the sensitive equipment, m2, at the frequency Za
coefficient c2 is made low, this effectively creates an anti-resonance condition in the transfer
admittance, so that the mass m2 becomes unresponsive to simple harmonic forcing at Z Za .
Therefore a vibration isolation effect can be achieved provided that the inertance b2 can be tuned so
as to match the antiresonance frequency to the excitation frequency. If a small scale mechanical system
is considered, then existing inerter designs probably cannot accomplish the task for reasons discussed
in Introduction. Therefore it is reasonable to attempt to synthesise the inerter effects by using the
feedback control loop shown schematically in Figure 1 (a). The purpose of the feedback loop is to78

generate a control force proportional to the relative acceleration between the equipment and the base,that is, to emulate effects of the inerter, b2, mounted in parallel to the suspension spring and damper,
shown in Figure 1 (b).
a)b)

Figure 1: The 2 DOF mechanical system equipped with a direct acceleration feedback loop, plot (a), withthe purpose of synthesizing the inerter, plot (b)The control force, f c , is applied by an electrodynamic actuator that reacts between the two masses .
The actuator is characterized by a transducer coefficient T, which is often referred to as voice coilconstant, inductance L, and the resistance R. The force generated by the actuator is proportional to theelectrical current flowing through the transducer coil through the transducer coefficient T:
f c Ti(1)
The current i, however, depends both on the voltage applied at the transducer electrical terminals, e, and on
the relative velocity, s x2  x1 , between its mechanical terminals, according to the following expression:eRi  sLi  sT x2  x1

(2)

where s iZ .
In the present controller scheme the voltage, e, is made proportional to subtracted outputs of thetwo accelerometer sensors through a voltage amplifier gain, g, Figure 1 (a). The error signals areprovided by two equal inertial accelerometers characterised by the transfer function Hs. This is thefrequency response function (FRF) between the accelerometer output and the true acceleration of astructure onto which the accelerometer is attached. The two accelerometers are assumed to be muchlighter than the sensitive equipment or the flexible base, so that their mechanical impedance can be
entirely neglected. Therefore the voltage at the actuator electrical terminals is given by the control
law:
e gH s s 2 x2  x1
(3)
The dynamics of the mechanical parts of the system, i.e. the system without the control loop
elements encircled by the red dashed line in Figure 1, can be represented by four mobility functions
Yi , j . They are the FRFs between the velocity of the mass i due to a force acting at the mass j. If i=j
the corresponding mobility is referred to as driving point mobility, otherwise it is referred to as atransfer mobility. By considering contributions of the primary and the control forces one can write:
sx1 Y1,1 f p  Y1,1 f c  Y1, 2 f c(4)79

Y2,1 f p  Y2,1 f c  Y2, 2 f c

sx2

(5)

By taking into account Eqs. (1-5) and the reciprocity principle that imposes Y2,1 Y1, 2 , the fully
coupled closed loop response of the system can be calculated in terms of five FRFs:
sgH s Y1,2  Y1,1 R  Ls

H e, f p

Y1,1  Y1,2 T  sgH s

Hi, f pH fc , f p(7)D1
TH i ,f p

(8)

Y1,2 2  Y1,1Y2,2 T 2  sgH s Y1,1Y2,2  Y1,2 2 T  Y1,1 R  Ls

H x1 , f p

(9)

D1 s

Y1,2 2  Y1,1Y2,2 T 2  sH s Y1,1Y2,2  Y1,2 2 T  Y1,2 R  Ls

H x2 , f p

where D1

(6)

D1

D1 s

(10)

2Y1,2  Y1,1  Y2,2 T 2  gH s Y1,1  2Y1,2  Y2,2 sT  R  Ls , H e , f p is the FRF between the actuator

voltage and the primary excitation force, H i , fp

is the FRF between the actuator current and theprimary excitation force, H f , f is the FRF between the control force and the primary excitation force,cp

H x1 , f p is the closed loop driving point receptance, and H x2 , f p is the closed loop transfer receptance.

The sensor-actuator open loop frequency response function can be obtained by calculating the FRF
between the subtracted accelerometer outputs and the voltage fed to the actuator in absence of the
primary excitation:
H s , a s 2 H s x2  x1
(11)
f 0
p

This can be done by substituting f p
H s,a0 into Eqs. (4,5) and taking into account also Eqs. (1-3):
sH sT 2Y1,2  Y2,2  Y1,1
2Y1,2  Y2,2  Y1,1 T 2  R  Ls

(12)

The transfer function Hs characterising the two accelerometer sensors can be written as (15):Hs

Z A2
Z A 2  2] A Z A s  s 2(13)

where Z A is the mounted natural frequency of the accelerometer, and ] A is the accelerometer damping
ratio. For simplicity, it is assumed that the accelerometer sensitivity is absorbed in the feedback gain g. In
other words, the accelerometer transfer function has been normalised so as to have a unit sensitivity for static
accelerations.
The four mechanical mobility functions Yi , j are, (14):
Y1,1
Y2,1
Y2, 2

where D2

80s s 2 m2  sc2  k 2
D2
Y1,2

s sc2  k2
D2

s s 2 m1  c1  c2 s  k1  k 2
D2

s 4 m1m2  c1  c2 m2  m1c2 s3  k1  k2 m2  m1k2  c1c2 s 2  k1c2  c1k2 s  k1k2 .

(14)
(15)
(16)

3. DISCUSSION
In order to assess the stability of the feedback loop, Nyquist criterion is used. The sensor-actuator
open-loop FRF, Eq. (12), is plotted using Bode and Nyquist diagrams for an example small scale
vibration isolation model problem. The properties of the system are given in Table 1.
Table 1 – Properties of the example system
ParameterValue

m1 (kg)0.1

m2 (kg)0.05

k1 (N/m)

1000

k 2 (N/m)

500

c1 (Ns/m)

1c2 (Ns/m)

1

T (N/A,Vs/m)

0.45

6

L ( 10 H)

63

R (: )

1.5

The properties of the actuator correspond to an off-the-shelf miniature moving coil linear motor(16). The properties of the two accelerometers are varied in order to illustrate their influence on thestability of the control loop. A situation if first considered assuming a standard, general purpose
piezoelectric accelerometer whose natural frequency and damping ratio have been identified bymatching the sensitivity function, Hs, Eq. (13), with that from its datasheet (17). The match can beseen in Figure 2, and the identified values of the mounted natural frequency and the damping ratio
are: f A 2SZ A 42 kHz and ] A 0.0158 . The agreement between the model results and the measured
data is very good so the simple accelerometer model given by Eq. (13) is valid.
(a)

(b)

Figure 2: The transfer function, Hs, of the two accelerometer sensors, plot (a): model, plot (b): measured
However, the stability analysis indicates that such an accelerometer is not appropriate for the task
considered in this study. As shown by Bode plot of the open loop sensor-actuator FRF in Figure 3 (a),the combination of the gradual phase lag of 45 degrees due to the inductance of the actuator coil andthe abrupt 180 degree phase lag due to the second order accelerometer dynamics , causes the locus of
the FRF to cross the negative real axis in the Nyquist plot, shown in Figure 3 (b). Furthermore, theamplitude at the crossover frequency of 42 kHz (the accelerometer mounted resonance frequency) isvery large due to the low accelerometer damping ] A . The amplitudes of the sensor-actuator openloop FRF at the two resonances due to the mechanical system response are in fact significantly lower81

that that due to the accelerometer resonance. As a result, no significant gain can be implemented
assuming standard gain margins and no significant inertance can be synthesised.
(a)

(b)

Figure 3: (a) Bode and (b) Nyquist plots of the open loop sensor-actuator FRF, H s , a , with f A]A

42 kHz and0.0158 .

A situation in which the accelerometer damping ratio is increased to its critical value ] A 1 withunchanged mounted resonance frequency is considered next. In this case the feedback gain can be set
to g= 0.43 V/V with a gain margin of 6 dB, as can be seen in the zoomed area of plot (b) of Figure 4,
since the locus crosses the negative real axis with the amplitude of 0.5. (The open loop sensor –actuator FRF has been multiplied by the feedback gain).
(a)

(b)

Figure 4: (a) Bode and (b) Nyquist plots of the open loop sensor-actuator FRF, H s , a , with f A]A

42 kHz and

1 , assuming the feedback gain of g= 0.43 V/VThis control loop synthesises the desired inertance, which can be seen in Figure 5, red solid line.
In fact, the figure shows the amplitude of the transfer receptance H x , f for four cases. The first case2p

is with the open loop system. In this case there can be seen two resonances due to the mechanicaldegrees of freedom and no antiresonances, as one might expect for transfer admittances in a 2 DOF
mechanical systems. However, if the feedback loop is closed with g= 0.43 V/V, ensuring a gain margin82

of 6 dB, then an antiresonance appears at the frequency of approximately 100 Hz. This corresponds
to a synthetized inertance of b2 k2 / ZA2 500 / (2S 100)2 kg 0.0012 kg . Therefore, at the frequencyof 100 Hz the mass m2 becomes rather unresponsive to the primary force fp exciting the mass m1 sothat the desired isolation effect is achieved.

Figure 5: The closed loop transfer receptance, H x , f , with no control (black faint line), and with control2pusing the feedback gains of 0.43 (red solid line), 0.45 (blue dashed line), and 1.24 (green dash-dotted line),that all ensure 6 dB gain margins, assuming the use of critically damped accelerometers having theirmounted natural frequencies of 42 kHz, 28 kHz, and 4.2 kHz, respectively, synthetizing the inertances ofapproximately 0.001 kg, 0.0035 kg, and 0.031 kg, respectively.
However, achieving the critical damping ratio in a miniature accelerometer system with a natural
frequency of 42 kHz is not trivial. This is because the accelerometer critical damping coefficient,increases with its natural frequency, cA,crit 2mAZA . As an illustration, if the accelerometer inertial
mass equals to mA 0.001 kg , then the required critical damping coefficient calculates to about 528
Ns/m. It is thus reasonable to consider an accelerometer with a lower natural frequency. Figure 5
shows that by reducing the natural frequency of the error accelerometers, it is possible to use higher
feedback gains and to synthetize larger inertances. The critically damped accelerometer in fact
behaves like a mechanical low-pass filter with a roll-off rate of 40 dB per decade above its natural
frequency, Figure 4 (a), green dashed line. This is beneficial in terms of the feedback loop stability
and the maximum admissible feedback gain, and thus, the maximum synthesizable inertance. However,the phase of the transfer function, Hs , is also altered by the filtering, and it lags by 90 degrees at theaccelerometer resonance frequency. As a result, the Nyquist locus rotates clockwise in the real imaginary plane indicating that the feedback loop produces a combination of relative acceleration andrelative velocity feedback, Figure 4 (b). Therefore, by decreasing the natural frequency of thecritically damped accelerometers, the antiresonances due to the synthetized inertances becomecharacterised by increasing active damping effects and the vibration isolation objective may not be
fully achieved.

4. CONCLUSIONS
An active control system emulating the effects of inerter element for vibration isolation isconsidered. It is in principle possible to apply direct relative acceleration feedback using inertial
accelerometers and electrodynamic reactive actuators. For stability reasons it is necessary to use

83highly damped accelerometers that effectively filter out the error signal above their natural frequency.
In particular, by using critically damped accelerometers with a high natural frequency the desired
inertance can be emulated by the control system. However, achieving large damping ratio s in smallscale accelerometers with a high natural frequency is problematic. In accelerometers with a low
natural frequency achieving high damping ratios is technically less challen ging, but in such a case thenatural frequency must not be too low in order to avoid the feedback loop delivering a combinationof active inertia and active damping which compromises the vibration isolation effect.

ACKNOWLEDGEMENTS
This project has received funding from the European Union's Horizon 2020 research andinnovation programme under the Marie Sklodowska-Curie grant agreement no. 657539.

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