© The Author(s), Published by EDP Sciences, 2024

1 Introduction

Inerter is a one-port, two-terminal element in mechanical networks which resists relative acceleration across its two terminals. The coefficient of this resistance, the inertance, is measured in kilograms. Inerter therefore behaves like a “relative mass”. The concept of relative mass was firstly introduced by Schönfeld [1], who mentioned the possibility of a two-terminal mechanical inertance and gave a rudimentary scheme of a physical realisation of the concept. Inerter stores kinetic energy proportional to its inertance multiplied by the square of relative velocity between its two terminals.

Inerters are especially interesting from the theoretical perspective of mechanical network analysis and synthesis. This is because they fill an empty niche enabling a complete analogy between mechanical and electrical networks. Assuming the force-current analogy, the electrical analogue of the inerter is the capacitor [1, 2]. 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 for elements like springs, dampers, inductances or resistors in lumped parameter mechanical or electrical networks [3]. However, a realistic element, for example a helical spring, can itself exhibit a rich dynamic behaviour aside from its stiffness in one direction [4]. This is equally true for other elements of lumped parameter mechanical networks, including the inerter.

Typical mechanical inerter designs include rack-and-pinion inerters [5], ball-screw inerters [6], and helical fluid channel inerters [7, 8]. Effects such as friction, stick-slip of the gear pairs, or the elasticity of the gears and connecting rods are inevitably present in gear-train inerter constructions. With helical fluid channel inerter, an inertance-like behaviour can be accomplished through accelerating hydraulic fluid back and forth into circular motion in the helical pipe. Such fluid-based inerters are characterised by a relatively large parasitic non-linear damping [7].

One of the most important characteristics of any physical realisation of the inerter is the ratio of its inertance to its mass. This ratio is normally required to be large to enhance the inertia effects of lightweight structures onto which the inerter is attached without significantly increasing their mass. In mechanical gear-train inerter designs the inertance can be several hundred times the mass of the inerter [5]. However, such inerters are typically mid-to large-scale and are not suitable for vibration control purposes in small-scale applications due to their large dimensions and large stroke.

Another class of inerter realisations are the electromechanical inerters. In these systems electromechanical transducers are shunted with appropriate electrical impedances at their electrical port in order to generate inertance-like effects at their mechanical port. Small-scale electromechanical transducers are characterised by a relatively low energy conversion efficiency [9], so it is necessary to use non-Foster shunt circuits in order to compensate for losses in the transducers [10]. 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 realise an ideal inerter element connected to additional elements that occur as a side-effect of using a particular shunting technique. For example, by shunting an electrodynamic (voice-coil) transducer with a particular negative impedance electrical circuit, inerter connected in series with a parallel spring damper-pair can be realised [10]. Another type of mechatronic inerter utilizes a rotary DC motor shunted with an appropriate electrical circuit in order to enhance the existing mechanical inertia effects with additional inertia generated via the electrical shunt [11]. Further active approaches to realise the inerter include the force feedback approach [12, 13]. Here a pair of collocated reactive actuators and a force sensor are used to feed back the output of the force sensor through both single and double integrators to drive the actuator. In such a way the inerter can be realised which is connected in series with a damper [13].

Inerter devices were recognized to be beneficial in a wide range of applications, from vibration control to new actuator concepts [3, 14]. For example, the relative inertia effect shifts the fundamental resonance frequency of inertial actuators to lower values without reducing their static stiffness [3, 14]. This is highly beneficial if they are used within a feedback vibration control loop, see [15, 16], and the references therein. A notable efficiency of inerters was demonstrated in vibration absorption problems [15, 17], where the goal is to reduce vibrations of a primary mass under forcing by adding a carefully suspended secondary mass [18–21]. In [22], two different positionings of the inerter device, grounded and inserted layouts, were compared in order to control vibrations of the primary system in a double tuned mass-damper-inerter structure.

A great deal of research has been dedicated to the optimisation of structures incorporating inerter elements. For example, a study on the optimisation of the dynamic behaviour of helical fluid channel inerters can be found in reference [23]. In reference [7], the authors consider an interesting concept for optimised structural control of buildings using inerter. Similarly, the authors in [24] are focused on H_∞ optimization with the similar goal of synthesizing an optimally designed Tuned Mass-Damper-Inerter for structures with model uncertainty.

In vibration isolation problems, an object, e.g. a piece of equipment, is isolated from the source of vibrations, or the other way around, a vibration source is isolated from the rest of the structure. In passive vibration isolation systems the two approaches are equivalent due to the reciprocity principle. With active vibration isolation systems this is not necessarily the case because the principle of reciprocity may not hold [25–28].

Dedicated vibration isolation systems may be needed in cases of excessive vibrations of rotating machinery [29], e.g. oscillatory forces from an engine in a car, which can propagate through the supporting structure, i.e. the car bodyworks [30], generating strong tonal sound and vibration components in the passenger area. On the other hand, road roughness, random bumps and holes create stochastic excitations that propagate through the tyres, suspension and chassis, and cause broad frequency band interior vibration and noise, causing passenger discomfort. In these situations, active vibration control can be considered to improve NVH characteristics of the vehicle [31]. It is often the case that passive, rubber-based isolation mounts in vehicles exhibit nonlinear effects which must be considered when designing active vibration isolation systems. In this sense, recently developed methodologies using neural network-based active controllers have shown promising results [32].

When it comes to the use of the inerter in vibration isolation problems it is in a way an analogue of using the Tuned Vibration Absorber (TVA) in vibration absorption problems. In particular, the TVA generates an antiresonance in the driving point receptance/mobility/accelerance of a primary oscillator equipped with the TVA. The frequency of this antiresonance is proportional to the square root of stiffness of the TVA suspension divided by the TVA mass [33, 34]. An inerter mounted in parallel with a classical parallel spring-damper isolator generates an antiresonance in the transfer receptance/mobility/accelerance between the source body forcing and the receiving body response. The frequency of this antiresonance is proportional to the square root of the stiffness of the existing spring divided by the inertance of the inerter [25]. This is quite different form the inertial vibration isolators which typically introduce an additional mass carefully suspended onto the base or the sensitive equipment including an additional spring.

The antiresonance can be used to generate a vibration isolation effect and is the principal approach to vibration isolation in this study. The inerter is emulated by using direct acceleration feedback control system with the aim of creating a tuneable inertance. By varying the inertance, the frequency of the above described antiresonance in the response of the receiving body can be tuned to track a possible variation of the frequency of the simple harmonic forcing of the source body. In the proposed acceleration feedback scheme, 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 to the 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 used. A fully coupled electromechanical model of a two Degree-Of-Freedom (2DOF) mechanical system equipped with the described active control loop is formulated. 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 realisable inertances that could be used to isolate simple harmonic vibrations. Suggestions for the active isolator design improvements in terms of the closed-loop stability are given. Therefore, a small-scale active realization of inerter with tuneable inertance relatively large with respect to its mass is proposed in order to isolate narrow frequency band vibration transmission from the source body to the receiving body.

The paper is structured as follows. In Section 2, a theoretical model encompassing a fully coupled lumped parameter system with two mechanical degrees of freedom is formulated. The control scheme resembling an inerter is proposed and a detailed theoretical investigation of closed-loop stability and performance is conducted. In Section 3, theoretical considerations are validated using a dedicated 3D-printed experimental test rig. Significant vibration isolation effect is recorded experimentally. A very good agreement with theoretical findings is observed.

2 Theoretical considerations

2.1 Mathematical model with ideal transducers

The mechanical system considered is a lumped parameter 2DOF system shown in Figure 1a. The system is composed of two oscillators, (m₁, k₁) and (m₂, k₂) elastically coupled with a stiffness k₁₂. This system is a simplified representation of a vibration isolation problem in which the receiving body, characterized by mass m₂ and stiffness k₂, is suspended with a spring k₁₂ and a damper c₁₂ onto a flexible base characterized by the source body, mass m₁ and stiffness k₁. Damper c₁₂ is to account for the dissipative losses caused by the air medium being pushed through the voice-coil actuator, as well as structural damping.

Figure 1 The 2DOF mechanical system equipped with a direct acceleration feedback loop, plot (a), with the purpose of synthesizing the inerter, plot (b).

Figure 2 Nyquist plot of the open-loop sensor-actuator transfer function assuming the transducers are ideal. The contour never crosses the negative real axis meaning that the system is stable for all gains g of the feedback loop.

The dynamic excitation is applied to the base mass through the simple harmonic primary force f_p1. The formulation presented considers time-harmonic functions, which are defined in complex form f(t) = Re(f(ω)e^iωt), where f(ω) is the complex amplitude, ω is the circular frequency and $i=\sqrt{-1}$ is the imaginary unit. As normally done in vibration studies, the formulation that follows thus refers to the complex amplitudes f(ω) of the time-harmonic functions f(t) and, for brevity, the frequency dependence is omitted.

With a goal of reducing the vibration transmission from the source mass m₁ to the receiving mass m₂, an inerter device (coloured in red in Fig. 1a) is added between the two masses. It is emulated by the control force f_c with its two components, an active component f_c1 and a reactive component, f_c2 oriented in opposite directions (see Fig. 1b).

If the sensors and actuators are considered ideal, this control force is proportional to the feedback gain g and the relative acceleration between the two masses:

$$f_{c1}\left(t\right)=-f_{c2}\left(t\right)=g\left({\ddot{x}}_2\left(t\right)-{\ddot{x}}_1\left(t\right)\right).$$(1)

The inertance b₁₂ is a measure of proportionality between the inerter force and the relative acceleration between two masses. From equation (1), this means that the feedback gain is directly equal to the inertance, i.e. b₁₂ = g, in this idealised case.

As discussed in, for example, reference [25], with the inclusion of the inerter of inertance b₁₂ into the suspension in Figure 1a, an antiresonance can be assigned to the transfer admittance between the primary force, f_p1, and the displacement/velocity/acceleration of the sensitive equipment, m₂ (see Fig. 3). The frequency of that antiresonance is given by ${\omega }_a=\sqrt{k_{12}{b}_{12}}$, where subscript a denotes the antiresonance [25]. If the damping coefficient c₁₂ is made low, this creates a sharp dip in the transfer admittance amplitude and the corresponding 90° phase lead, so that the mass m₂ becomes unresponsive to simple harmonic forcing at ω = ω_a. Therefore, a vibration isolation effect can be achieved provided that the inertance b₁₂ can be tuned to match the antiresonance frequency ω_a to the frequency of the primary forcing f_p1. If a small-scale mechanical system is considered, then existing inerter designs probably cannot accomplish the task for reasons discussed in the Introduction. Therefore, in this paper, a scheme to emulate the inerter effects by using the feedback control loop shown in Figure 1b is proposed. The purpose of the feedback loop is to generate a control force proportional to the relative acceleration between the equipment and the base, that is, to mimic effects of the inerter, b₁₂, mounted in parallel to the suspension spring and damper, shown in Figure 1a.

Figure 3 Closed-loop transfer function for different values of relative acceleration feedback.

The stability of the closed-loop system can be analysed through the Nyquist stability criterion of the sensor-actuator transfer function. Here, “sensor” in fact corresponds to the difference of the outputs of two accelerometers mounted on masses m₁ and m₂. Therefore, the open-loop transfer function is given by

$$H_{\mathrm{ol}}=\frac{s^2\left(x_2-x_1\right)}{f_c},$$(2)

where the f_c is the control force exerted on the system by the control actuator. As the system is assumed linear, both displacements x₁ and x₂ are given by the superimposed effects of the two control force components:

$$x_1=X_{1,1}{f}_{c1}+X_{1,2}{f}_{c2},$$(3)

$$x_2=X_{2,1}{f}_{c1}+X_{2,2}{f}_{c2}.$$(4)

The four receptance functions X_i,j, where i, j = 1, 2, connect two points of excitation j to the corresponding displacement response i. They are the frequency response functions (FRFs) between the displacement of the mass i due to a force acting at the mass j. If i = j, the corresponding receptance is referred to as driving point receptance, otherwise it is referred to as a transfer receptance. The receptance functions may be calculated through the dynamic stiffness matrix

$$\mathbf{S}=\mathbf{M}{s}^2+\mathbf{C}s+\mathbf{K},$$(5)

where M is the mass matrix, C is the damping matrix, and K is the stiffness matrix of the system:

$$\mathbf{M}=\left[\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right],$$(6)

$$\mathbf{C}=\left[\begin{array}{cc}{c}_{12}& -c_{12}\\ -c_{12}& c_{12}\end{array}\right],$$(7)

$$\mathbf{K}=\left[\begin{array}{cc}{k}_1+k_{12}& -k_{12}\\ {-k}_{12}& k_2+k_2\end{array}\right],$$(8)

such that

$$\mathbf{Sx}\left(s\right)=\mathbf{f}\left(s\right),$$(9)

so that the receptance matrix becomes

$$\mathbf{X}={\mathbf{S}}^{-1}=\left[\begin{array}{cc}{X}_{1,1}& X_{1,2}\\ X_{2,1}& X_{2,2}\end{array}\right].$$(10)

The question is, for what gain g does the contour gH_ol cross the Nyquist point −1 + 0i. In Figure 2, two contours for two different feedback gains are shown. It is visible that the Nyquist contour never crosses the negative real axis, meaning that the Nyquist point −1 + 0i, is never encircled, no matter how big feedback gain g becomes. Therefore, the closed-loop system is unconditionally stable. This is only because ideal sensors and actuators are assumed.

The closed-loop accelerance response H_2,1 = s²x₂/f_p1 that determines the acceleration of mass m₂ due to an excitation f_p1, is again calculated as a superposition of all the forces contributing to the closed-loop response:

$$x_2=X_{2,1}{f}_{p1}+X_{2,1}{f}_{c1}+X_{2,2}{f}_{c2},$$(11)

$$x_1=X_{1,1}{f}_{p1}+X_{1,1}{f}_{c1}+X_{1,2}{f}_{c2}.$$(12)

This in turn defines the closed-loop transfer accelerance $H_{2,1}=\frac{s^2{x}_2}{f_{p1}}$ as

$$H_{2,1}=\frac{s^2\left(g{s}^2\left(X_{2,2}{X}_{1,1}-X_{2,1}^2\right)+X_{2,1}\right)}{1+g\left(X_{1,1}-{2X}_{2,1}+X_{2,2}\right)s^2}.$$(13)

If g = 0, H_2,1 = s²x_2,1 which is the transfer accelerance of the passive system. Figure 3 shows the amplitudes of the different closed-loop accelerances H_2,1 for different feedback gains g. Red dashed line (g₁) corresponds to the passive system where the feedback gain is zero. Two resonant frequencies of the 2DOF lumped parameter system are evident. As the feedback gain increases, an antiresonance begins to appear at higher frequencies first. With further increasing the feedback gain, the antiresonance gradually shifts to lover frequencies, corresponding to relation ${\omega }_a=\sqrt{k_{12}{b}_{12}}=\sqrt{k_{12}g}$, which is the zero of transfer function in equation (13). The parameters of the lumper-parameter system are shown in Table 1.

2.2 Mathematical model with practical transducers

It was shown in the previous section that unconditional stability of the 2DOF lumped parameter system with the inerter device inserted is guaranteed if the transducers are ideal. This is because this control scheme resembles a collocated control feedback. However, this is not to be expected in real-world because the transducer dynamics is inevitably going to impact the closed-loop response. In the following step, the mathematical model introduced in Section 2.1 is extended with a detailed mathematical model of actuators and sensors.

The actuators used are off-the-shelf miniature moving coil linear motors (voice-coil actuators). Their electrodynamic properties can be described by a transducer coefficient T, which is often referred to as voice coil constant, inductance L, and the resistance R (see Fig. 1b). The force generated by the actuator is proportional to the electrical current flowing through the transducer coil via the transducer coefficient T. Therefore, the control force is given by:

$$f_c = -T\bullet i_c.$$(14)

The current i_c, however, depends both on the voltage applied to the transducer electrical terminals, e_c, and on the relative velocity, s(x₂ − x₁) between its mechanical terminals, according to the following expression:

$$e_c= R{i}_c+\mathrm{sL}{i}_c+\mathrm{sT}\left(x_2-x_1\right).$$(15)

Note that all the sensors and all the transducers are assumed identical in terms of their parameters, so it also means that the voltage at the primary excitation actuators is:

$$e_{p1}= R{i}_{p1}+\mathrm{sL}{i}_{p1}+\mathrm{sT}{x}_1.$$(16)

In the present controller scheme, the control voltage e_c is made proportional to subtracted outputs of the two accelerometer sensors through a voltage amplifier gain, g* (see Fig. 1b). In other words, a voltage command is used. The error signals are provided by two equal inertial accelerometers characterised by the transfer function H_s. This is the FRF between the accelerometer output and the true acceleration of a structure onto which the accelerometer is attached. The moving masses of the two accelerometers are assumed to be much smaller than the sensitive equipment (m₂) or the flexible base (m₁), so that their mechanical impedance can be entirely neglected. The voltage at the actuator electrical terminals is then given by the control law:

$$e_c = g^{*}{H}_s{s}^2\left(x_2 -{ x}_1\right).$$(17)

The asterisk is to differentiate the gain in equation (17) which is in Vs²/m with the one when transducers are ideal, and the gain is equal to inertance in kilograms. 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 1b, can be represented by four receptance functions X_i,j from equation (10). By considering again equations (11), (12) as well as equations (14)–(17), and the reciprocity principle that imposes X_1,2 = X_2,1, the fully coupled closed-loop response of the system can be calculated in terms of five FRFs:

$$H_{e_c,e_{p1}}=\frac{g^{*}{H}_s{s}^2 \left(X_{1,1}-X_{2,1}\right)\bullet \left(R+L\bullet s\right)T}{D_1},$$(18)

$$H_{i_c,e_{p1}}=\frac{-\left(s\left(X_{1,1}-X_{2,1}\right)\left(T-H_s{g}^{*}s\right)T\right)}{D_1},$$(19)

$$H_{f_c,e_{p1}}=\frac{-T^{2}s\left(X_{1,1}-X_{2,1}\right)\left(-H_s{g}^{*}s+T\right)}{D_1},$$(20)

$$H_{{\ddot{x}}_1,e_{p1}}=\frac{-s^2\left(T\left(T{g}^{*}{H}_s\left(X_{1,1}{X}_{2,2}{{-X}_{2,1}}^2\right)s^2+\left(\left(X_{2,1}^2-X_{1,1}{X}_{2,2}\right)T^2+{\mathrm{LX}}_{1,1}\right)s+X_{1,1}R\right)\right)}{D_1},$$(21)

$$H_{{\ddot{x}}_2,e_{p1}}=\frac{-s^2\left(T\left(T{g}^{*}{H}_s\left(X_{1,1}{X}_{2,2}{{-X}_{2,1}}^2\right)s^2+\left(\left(X_{2,1}^2-X_{1,1}{X}_{2,2}\right)T^2+{\mathrm{LX}}_{2,1}\right)s+X_{2,1}R\right)\right)}{D_1},$$(22)

where

$$D_1=g^{*}{\mathrm{TH}}_s\left(\left(X_{2,1}^2-X_{2,2}{X}_{1,1}\right)T^2+L\left(X_{2,2}-{2X}_{2,1}+X_{1,1}\right)\right)s^3+\left(\left(X_{2,2}{X}_{1,1}-X_{2,1}^2\right)T^2-2L\left(X_{1,1}-X_{2,1}+\frac{1}{2}{X}_{2,2}\right)T^2+R{g}^{*}{H}_{s}T\left(X_{2,2}-{2X}_{2,1}+X_{1,1}\right)+L^2\right)s^2+2\left(\left(X_{2,1}-X_{1,1}-\frac{1}{2}{X}_{2,2}\right)T^2+L\right)\mathrm{Rs}+R^2.$$(23)

$H_{e_{c1},e_{p1}}$ is the FRF between the control actuator voltage and the primary excitation voltage, $H_{i_c,e_{p1}}$ is the FRF between the actuator current and the primary excitation voltage, $H_{f_c,e_{p1}}$ is the FRF between the control force and the primary excitation voltage, $H_{{\ddot{x}}_1,e_{p1}}$ is the closed loop acceleration of the first mass m₁ due to the primary actuator voltage, and $H_{{\ddot{x}}_2,e_{p1}}$ is the closed loop acceleration of the second mass m₂ due to the primary actuator voltage.

The sensor-actuator open loop FRF can be obtained by calculating the FRF between the subtracted accelerometer outputs and the voltage fed to the control actuator in absence of the primary excitation:

$${H_{s,a}=s^2{H}_s\left(x_2-x_1\right)|}_{f_{p1}=0}.$$(24)

This can be done by substituting f_p1 = 0 into equations (11) and (12) and also considering equations (17) and (21):

$$H_{s,a}=\frac{s^2{H}_{s}T\left({2X}_{1,2}-X_{2,2}-X_{1,1}\right)}{s\left(2X_{1,2}-X_{1,1}-X_{2,2}\right)\bullet T^2+R+\mathrm{Ls}}.$$(25)

The transfer function H_S characterising the two accelerometer sensors can be written as [35]:

$$H_S=\frac{{\omega }_A^2}{{\omega }_A^2+2{\zeta }_A{\omega }_{A}s+s^2},$$(26)

where ω_A is the mounted natural frequency of the accelerometer, not to be confused with the frequency of the antiresonance ω_a. ζ_A is the accelerometer damping ratio, see Table 2. 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 to have a unit sensitivity for static accelerations.

Equation (25) is thus the open-loop sensor-actuator FRF used in the following section to study the stability of the active system.

2.3 Stability

Although the control approach is physically well-founded, its applicability is not straightforward. For example, the relative acceleration sensor is collocated with the action-reaction force actuator, but the two transducers are not dual [36, 37], i.e., they are not complementary in terms of mechanical power. As a result, the inherent frequency response of the sensor-actuator transducers can inhibit the stability of the feedback loop, as discussed in references [38, 39]. In order to assess the stability of the feedback loop, Nyquist criterion is again used. The sensor-actuator open-loop FRF, equation (25), is first analyzed for the same example small-scale vibration isolation problem as in Section 2.1. The mechanical parameters are set to be identical to the previous considerations assuming ideal transducers. The properties of the sensors and the actuators are given in Table 2.

Figure 4 shows the Nyquist plot of the sensor-actuator open-loop FRF that includes the transducer dynamics. The contour for two feedback gains is given – a stable and unstable. Although the locus is predominantly in the positive imaginary two quadrants of the complex plane, as one would expect from a collocated force-acceleration transducer system, it nevertheless crosses towards the negative imaginary quadrants with a crossover of the negative real axis.

Figure 4 The Nyquist plot of the sensor actuator open-loop FRF Hs,a obtained theoretically with transducer dynamics included in the feedback loop.

This crossover occurs near the natural frequency of the accelerometer transducers. This suggests that a larger damping ratio ζ_A of the accelerometer might increase stability margins of the closed-loop system. Furthermore, due to the combined effects of the voltage command approach and the finitely small inductance of the electrodynamic actuator the locus is slightly rotated in the clockwise direction. The voltage command applied to the voice-coil actuator effectively generates a low-pass filter with a cut-off frequency of about 3.8 kHz. This cut-off frequency is the frequency of the pole of the R-L electrical subsystem with values like in Table 2.

Since off-the-shelf accelerometers are used, it is difficult to intervene in their damping ratio. Instead, in order to increase the stability margin of the open-loop sensor actuator transfer function, an additional simple first-order Butterworth low-pass filter is added in the feedback loop, with a cut-off frequency of 1000 Hz. The filter affects the open-loop response in two ways: 1) the very high frequency sensor resonance (42 kHz) in the open-loop sensor-actuator FRF becomes attenuated by the 20 dB/decade roll-off of the filter, significantly decreasing the amplitude of the FRF at the crossover frequency; 2) the phase shift causes the contour to additionally rotate in clockwise direction, thereby increasing the phase margin. The effect of the low-pass filter is shown in Figure 5, where the open-loop response which would otherwise make the closed-loop response unstable (blue dashed line) is now stabilized by the filter (Fig. 5c), effectively producing a phase margin of around φ₂₁ = 14°. The additional decrease of the filter cut-off frequency would furthermore rotate the contour in the clockwise direction. Note however that such a decrease would cause the filter to act like an integrator, turning the acceleration feedback into velocity feedback. This is visible in the closed-loop response obtained experimentally, where a damping effect in the antiresonance appeared, see Figure 7b, yellow dash-dotted line. In conclusion, the system can be made stable if the feedback gain is below a certain value (about 0.18 Vs²/m) but will go unstable if the feedback gain is increased above this threshold.

Figure 5 The Nyquist plot of the sensor actuator open-loop FRF obtained theoretically (a, c) and experimentally (b, d), with (c, d) and without (a, b) low-pass filter included in the feedback loop. Clockwise rotation of the contour because of the additional low-pass filter is evident. Phase margins are as follows: (a) limit of stability, φ11 = 0; (c) φ21 = 14°; (b) φ12 = 5.95°; (d) φ22 = 10.91°.

Figure 6 The transfer accelerance of the active system with increasing feedback gains: (a) up to 10 000 Hz where the accelerometer resonance is visible, (b) up to 800 Hz to capture the main structural modes of the system.

Figure 7 The amplitude and phase of the FRF of the active system with low-pass filter included: (a) obtained theoretically, (b) obtained experimentally. The peak at 50 Hz is due to electrical network noise.

2.4 Performance

2.4.1 Without low-pass filter

When the active control is switched on, the transfer accelerance between the primary forcing of mass m₁ and the acceleration of mass m₂ becomes characterized by the antiresonance which can be tuned by increasing the feedback gain until instability occurs. Figure 6 shows the amplitude of the FRF between the primary actuator voltage e_p1 and acceleration of mass m₂ obtained theoretically (see Eq. (24)). In contrast to the response when transducers were ideal, a 42 kHz resonance of the accelerometer appears.

As the feedback gain increases, the antiresonance frequency shifts downwards with the lowest obtainable frequency of about 220 Hz due to the stability limit. Because of the accelerometer resonance, the stability margin is relatively low, and the antiresonance does not enter the frequency range between the two resonances because the system goes unstable for relatively small feedback gains.

2.4.2 With low-pass filter

Figure 7 shows the closed loop response with the low-pass filter inserted in the feedback loop. This is modeled mathematically by augmenting the feedback law equation (24) that now takes the form:

$$e_c = g^{*}{G}_{\mathrm{LP}}{H}_s{s}^2\left(x_2 -{ x}_1\right),$$(27)

where

$$G_{\mathrm{LP}}=\frac{1}{1+\tau s},$$(28)

and $\tau =\frac{1}{2\pi f_{\mathrm{cut}-\mathrm{off}}}$, where f_cut-off = 1000 Hz.

As the feedback gain increases, the antiresonance frequency shifts downwards with the lowest obtainable frequency of about 90 Hz, again due to the stability limit. This antiresonance frequency indicates the actively realised inertance of about 57 g given that $b_{12}=\frac{k_{12}}{{{\omega }_a}^2}$, see reference [25]. This is about ten time the mass of the actuator (coil + magnet), see Table 2. Note the reduced amplitude roll-off of the closed-loop acceleration response when compared to the passive case. This is because the generated inertance effect between the two bodies at higher frequencies prevails and locks the two bodies together causing them to behave as one body having a total mass of (m₁m₂ + b(m₁ + m₂))/b.

Because the transducers are no longer ideal, the inertance is no longer equal to the feedback gain. In fact, the inertance synthesised in the present way can be represented by a complex frequency dependant function characterised by its own amplitude and phase. By defining a complex synthetized inertance as the ratio between the control force and the relative acceleration of the two actuator terminals, one obtains

$$b_{12}^{*}=\frac{f_c}{s^2\left(x_2-x_1\right)},$$(29)

which, after using equations (14)–(17) becomes

$$b_{12}^{*}=\frac{T\left(T-\mathrm{gs}{G}_{\mathrm{LP}}{H}_s\right)}{s\left(R+\mathrm{sL}\right)},$$(30)

where asterisk is used in equations (29) and (30) to distinguish the complex frequency dependent synthesized inertance from the passive inertance shown in Figure 1a.

Note in equation (30) that the synthesized inertance does not depend on mechanical parameters of the system, it is only a function of the properties of the sensors, actuators and the low-pass filter. These can, in principle, be chosen according to a particular application. The amplitude and phase of the complex inertance $b_{12}^{*}$ are shown in Figure 8.

Figure 8 Amplitude-frequency and phase-frequency characteristics of inertance with included transducer dynamics: (a) amplitude-frequency, (b) phase-frequency.

Figure 9 Relation between the modulus of the complex inertance and feedback gain with included transducer dynamics for different frequencies: (a) up to inertance of 0.2 kg, (b) magnified for low values of the feedback gain up to 0.01 kg.

The amplitude remains approximately constant between 2 Hz and 200 Hz but starts to drop as the excitation frequency increases. Above 10 000 Hz, the added mass is practically non-existent even for the largest feedback-gain for which the closed-loop system is stable. In this frequency band, the frequencies are above the cut-off frequency of the filter, which practically acts as an integrator. This turns acceleration feedback into velocity feedback, and so the feedback loop acts as a damper (see Fig. 8a). Furthermore, there is an apparent phase-lag of the output force in comparison to the relative acceleration. The phase lag is limited at frequencies below 100 Hz but increases at higher frequencies until the accelerometer resonance frequency, where if drops significantly (see Fig. 8b).

Note that there is a frequency band for which the complex inertance is not zero when the feedback gain is zero, and the system is passive. This is shown by the red solid line in Figure 8. From equation (30), for g^* = 0,

$$b_{12}^{*}=\frac{T^2}{s\left(R+\mathrm{sL}\right)}.$$(31)

In this regime, the control voltage e_c = 0 so the control actuator becomes short-circuited. This means that the voltage equation (15) becomes

$$0= R{i}_c+\mathrm{sL}{i}_c+\mathrm{sT}\left(x_2-x_1\right),$$(32)

so there is still non-zero control current and thus the non-zero control force. At low frequencies, the time-derivative of the current may be neglected, so the control force is approximately equal to

$$f_c=T{i}_c\approx \frac{s{T}^2\left(x_1-x_2\right)}{R}.$$(33)

This effectively resembles a damper with a damping factor of $\frac{T^2}{R}$. At higher frequencies the control force

$$f_c=T{i}_c\approx \frac{s{T}^2\left(x_1-x_2\right)}{\mathrm{sL}}$$(34)

is dominated by factor of sL. This means that the control force practically becomes proportional to the relative displacement, with the proportionality factor $\frac{T^2}{L}$. Therefore, for the short-circuited actuator (g* = 0), the control loop resembles the so called, relaxation isolator [40], whose mechanical representation is a spring-damper connected in series (see Fig. 8a).

Nonetheless, as the feedback gain increases, the relationship between inertance and feedback gain is fairly linear as shown in Figures 8a and 8b.

In order to investigate power consumption of the active system, the transfer function between control current and voltage (Eq. (19)) is considered next. Figure 10 shows the amplitudes of the FRFs of the control current and the acceleration of the receiving mass with respect to the primary actuator voltage. It is visible that they are characterized by different antiresonances, that is, the control current antiresonance appears at the frequency:

$${\omega }_{i_c}=\sqrt{\frac{k_2}{m_2}}.$$(35)

Figure 10 The amplitude of the FRF of current and acceleration of the receiving mass with respect to the voltage applied to the primary excitation actuator. The antiresonance frequency for the system at hand is $f_{i_c}={\omega }_{i_c}2\pi =12\pi \sqrt{k_2{m}_2}=61.5 \mathrm{Hz}$.

Figure 11 The experimental setup. Accelerometers measuring acceleration signal from both masses is depicted in blue colour, rigid masses and leaf springs are depicted in purple and green colour, respectively. Primary and control actuators are noted in blue. The second actuator is not used in the experimental analysis.

Figure 12 Scheme of the experimental setup for measuring the closed-loop response. The acceleration signal subtraction is performed using a simple operational amplifier based signal subtractor.

This is further elaborated in the Appendix A of the paper. Two additional frequencies near the antiresonance frequency of the transfer accelerance at 90 Hz are examined. The amplitudes of the FRF of the control current with respect to the primary voltage at 85 Hz, 90 Hz and 95 Hz are 0.301 A/V, 0.2897 A/V, and 0.3046 A/V respectively. If the voltage applied to the primary actuator is set to, for example, 0.4 V (identical to the experimentally obtained time response, see Fig. 13), then the amplitude of the current to the control actuator is around 0.12 A, which means that the power dissipated at the resistor in the control actuator circuit is $P=i_c^{2}R=\frac{1}{T}{\int }_0^T{i}_c^2 R \mathrm{d}t=0.0108 \mathrm{W}$, where R is the voice-coil resistance (see Fig. 1b). Therefore, for small-scale structures, dissipation of energy due to the resistor losses is not significant. However, for large scale oscillation control problems (for example active anti-roll tanks in ships), power consumption by the control system becomes a big limitation, see [41, 42].

Figure 13 Time domain velocity response of the second mass if the system is turned active in comparison to the passive case. The response is shown for three values of frequencies around the antiresonance frequency: (a) 90 Hz, (b) 85 Hz, (c) 95 Hz.

3 Experimental analysis

The experimental setup is shown in Figure 11. The mechanical part of the setup is composed of a peripheral rigid frame and two elastically coupled mechanical oscillators inside. Note that the setup mimics the lumped parameter model from Figure 1, as the masses are realised by concentrating fairly rigid lumps of material whereas the stiffnesses are realised by lightweight flexible straight or curved beam elements. The specific details on design and manufacturing of the experimental setup together with the technical scheme of the 3D-printed test rig are shown in Appendix B to the paper. Appendix B also lists high-order mode shapes and natural frequencies of the test rig obtained using Finite Element Method (FEM) analysis.

The stability of the active control system may again be analyzed by investigating the properties of the open-loop Nyquist locus of the sensor-actuator FRF. The Nyquist contour is obtained by exciting the system with a white noise voltage signal through the control actuator. The difference between the two acceleration signals is measured with piezoelectric accelerometers and subtracted to obtain the error signal. The Nyquist plots are presented in Figures 5b and 5d. It is evident that adding the low-pass filter to the feedback signal increased the phase margin from φ₁₂ = 5.95° to φ₂₂ = 10.91°. The shape of the locus is quite similar to that obtained theoretically so that the maximum stable feedback gain is limited. The clockwise rotation of the contour due to the low-pass filters is also visible.

Figure 7b shows the amplitude of the transfer accelerance of the active system obtained experimentally, which were first shown in reference [43].¹ The functions are obtained by exciting the left mass of the experimental rig in Figure 11 with the primary actuator on the left. The actuator is driven with a white noise voltage signal generated by the spectrum analyser, and the acceleration response of the second mass on the right is measured. Charge amplifier is used to apply the additional low-pass filter with the 1 kHz cut-off frequency. Signals from the two accelerometers are subtracted and amplified with the feedback gain through the power amplifier. The amplified signal is then fed back to the control actuator. This experimental setup is schematically shown in Figure 12.

In Figure 7b, a similar shift of the antiresonance frequency towards lower frequencies is visible when the feedback gain is increased from zero. The lowest antiresonance frequency is about 90 Hz and it corresponds to the inertance of about 57 g. This is about ten times the total mass of the voice coil actuator used to apply the control force (5.7 g).

Figure 13 shows the closed-loop response of mass m₂ in the time domain, for three frequency excitations near the lowest antiresonance frequency achieved. If the frequency of excitation exactly matches the antiresonance frequency of 90 Hz, the acceleration response is reduced by about 40 dB when compared to the passive case. This frequency, however, is very close to the first natural frequency of the passive system, so results in the time domain for additional 2 frequencies, 85 Hz and 95 Hz are shown at these frequencies the vibration isolation effect decreases the response in comparison to the passive system are about 18 and 26 dB, respectively.

4 Conclusions

Inerter element is realised by means of relative acceleration feedback. Although the control approach is physically well-founded, its application is not straightforward, mainly for the feedback loop stability problems. This is because although the relative acceleration sensor is collocated to the reactive force actuator, the two transducers are not dual i.e., they are not complementary in terms of mechanical power, and the phase margin of the system is relatively low. Also, the inherent frequency response of the sensor-actuator transducers inhibits the stability of the feedback loop. This requires the use of two low-pass filters, one of which is intrinsic to the use of the voltage command to the voice coil actuator characterised by a resistance-inductance electrical cut-off frequency. Usable inertances can be realised which generate the desired antiresonance effect. The maximum inertance is about 57 g which is about 10 times the weight of the actuator (coil + magnet) used. The synthetized inertance, however, is a complex frequency dependent function. It is shown that it does not depend on the mechanical system properties, but only on the properties of the transducers used and the signal conditioning in the feedback loop. As a result, it can be adjusted to be predominantly real in the frequency range up to 1600 Hz such that the antiresonance can be generated, and its frequency tuned by varying the feedback gain. A significant vibration isolation effect can be generated in the narrow frequency band around the antiresonance.

Acknowledgments

This work has been supported by Croatian Science Foundation under the project IP-2019-04-5402 and IP-2022-10-4408. Goran Radan, dipl. ing. is gratefully acknowledged for designing and manufacturing the miniature power amplifier for driving the electrodynamic actuators.

Conflicts of interest

The authors disclose that they do not have a financial relationship with the organization that sponsored the research.

They also state that they have the full control of all primary data and that they agree to allow the journal to review their data if requested.

Data availability statement

The data are available from the corresponding author on request.

Appendix A

Here, an investigation of a lumped-parameter system with ideal inerter element is carried out, such as shown in Figure 1a.

The equations of motion are:

$$\left(m_1+b_{12}\right){\ddot{x}}_1-b_{12}{\ddot{x}}_2+c_{12}{\dot{x}}_1-c_{12}{\dot{x}}_2+\left(k_1+k_{12}\right)x_1-k_{12}{x}_2=f_{p1}$$(A1)

$${-b}_{12}{\ddot{x}}_1+\left(m_2+b_{12}\right){\ddot{x}}_2-c_{12}{\dot{x}}_1+c_{12}{\dot{x}}_2+\left(k_{12}+k_2\right)x_2-k_{12}{x}_1=0.$$(A2)

The equations of motion equations (A1) and (A2) can also be written in the matrix form as

$$\mathbf{F}=\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{Kx},$$(A3)

where C and K are identical as in equations (7) and (8), and inertance is added to the mass matrix M,

$$\mathbf{M}=\left[\begin{array}{cc}{m}_1+b_{12}& -b_{12}\\ -b_{12}& m_2+b_{12}\end{array}\right],$$(A4)

$$\mathbf{C}=\left[\begin{array}{cc}{c}_{12}& -c_{12}\\ -c_{12}& c_{12}\end{array}\right],$$(A5)

$$\mathbf{K}=\left[\begin{array}{cc}{k}_1+k_{12}& -k_{12}\\ {-k}_{12}& k_{12}+k_2\end{array}\right],$$(A6)

x(t), $\dot{\mathbf{x}}\left(t\right)$ and $\ddot{\mathbf{x}}\left(t\right)$ are the displacement, velocity and acceleration column vectors respectively, and F is excitation column vector,

$$\mathbf{x}=\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right],$$(A7)

$$\mathbf{F}=\left[\begin{array}{c}{f}_{p1}\left(t\right)\\ 0\end{array}\right].$$(A8)

Assuming a simple harmonic excitation and expressing the excitation and the steady-state response in the exponential form $\mathbf{F}\left(t\right)=\widehat{\mathbf{F}}{e}^{\mathrm{i\omega }t}$ and $\mathbf{x}\left(t\right)=\widehat{\mathbf{x}}{e}^{\mathrm{i\omega }t}$, where $i=\sqrt{-1}$, equation (A3) can be written as

$$\mathbf{S}\left(\mathrm{i\omega }\right)\mathbf{x}\left(\mathrm{i\omega }\right)=\mathbf{F}\left(\mathrm{i\omega }\right),$$(A9)

where S(iω) is the dynamic stiffness matrix as in equation (5). Solution of the equation (A9) is again obtained by inversion of the dynamic stiffness matrix, as

$$\mathbf{x}\left(\mathrm{i\omega }\right)={\mathbf{S}}^{-\mathbf{1}}\left(\mathrm{i\omega }\right)\mathbf{F}\left(\mathrm{i\omega }\right).$$(A10)

Differentiating equation (A10) two times in order to obtain accelerations results in expression

$${\left(\mathrm{i\omega }\right)}^2\mathbf{x}\left(\mathrm{i\omega }\right)=\mathbf{H}\left(\mathrm{i\omega }\right)\mathbf{F}\left(\mathrm{i\omega }\right),$$(A11)

where H(iω) = (iω)²S⁻¹ is the accelerance matrix representing four FRFs between the acceleration of each degree of freedom and the force applied to it. The steady-state complex acceleration response of mass m₂ on excitation input f_p1(iω) can now be expressed with H₂₁(iω) element of matrix H(iω),

$$H_{21}\left(\mathrm{i\omega }\right)=\frac{N_{H_{21}}\left(\mathrm{i\omega }\right)}{D\left(\mathrm{i\omega }\right)},$$(A12)

where $N_{H_{21}}\left(\mathrm{i\omega }\right)={\left(\mathrm{i\omega }\right)}^4{b}_{12}+{\left(\mathrm{i\omega }\right)}^3{c}_{12}+{\left(\mathrm{i\omega }\right)}^2{k}_{12}$, and $D\left(\mathrm{i\omega }\right)={\left(\mathrm{i\omega }\right)}^4\left(\left(b_{12}+m_2\right)m_1+b_{12}{m}_2\right)+{2\left(\mathrm{i\omega }\right)}^3{b}_{12}{c}_{12}+{\left(\mathrm{i\omega }\right)}^2\left(\left(b_{12}+m_2\right)k_1+\left(m_1+m_2\right)k_{12}+\left(m_1+b_{12}\right)k_2-{c_{12}}^2\right)+2\left(\mathrm{i\omega }\right)c_{12}{k}_{12}+\left(k_{12}+k_2\right)k_1+k_{12}{k}_2$. Note that the induced antiresonance frequency is defined as the null of the numerator, $N_{H_{21}}\left(\mathrm{i\omega }\right)=0$, assuming vanishing damping, which gives

$$\mathrm{AR}=\sqrt{\frac{k_{12}}{m_2}}.$$(A13)

Figure A1 shows the amplitude of system’s transfer accelerance H_2,1 for different inertances with respective high frequency asymptotes.

Figure A1 Amplitude of the FRF H2,1 with ideal inerter device, with highlighted high-frequency amplitude.

It can be seen that at high frequencies the non-zero inertance results in a locking effect which is a notable drawback in terms of vibration isolation relative to the passive case, should the primary excitation be broadband. Figure A2 shows how the high-frequency asymptote changes with the change of inertance b₁₂. The value of the asymptote can be calculated as the limit value of the transfer accelerance when frequency tends to infinity, $\underset{\omega \to \infty }{\lim }\left|H_{2,1}\right|$, which yields:

$$\mathrm{As}=\frac{b_{12}}{m_1{m}_2+b_{12}\left(m_1+m_2\right)}.$$(A14)

Figure A2 Shift of the high-frequency asymptote with respect to the inertance b12.

The asymptote has got its own asymptote, approaching the value 1/(m₁ + m₂) as b₁₂ increases towards infinity, see Figure A2.

Figure A3 shows the FRF of the inerter force with respect to the input excitation force. This FRF is characterized by an additional antiresonance corresponding to frequency

$$\omega =\sqrt{\frac{k_2}{m_2}},$$(A15)

which can be calculated by equating the numerator of

$$H_{f_b,f_{p1}}=\frac{N_{f_b,f_{p1}}\left(\mathrm{i\omega }\right)}{D\left(\mathrm{i\omega }\right)},$$(A16)

where $N_{f_b,f_{\mathrm{p}1}}\left(\mathrm{i\omega }\right)= {b_{12}{m}_2\left(\mathrm{i\omega }\right)}^4-{b_{12}{c}_{12}\left(\mathrm{i\omega }\right)}^3+{b_{12}{k}_2\left(\mathrm{i\omega }\right)}^2$ and D(iω) is identical to the one in equation (A12), to zero, under the vanishing damping assumption (c_1,2 = 0). This is exactly the cause of the antiresonance in corresponding control current FRF shown in Figure 10.

Figure A3 Amplitude of the FRF $H_{f_c,f_{p1}}$ with ideal inerter device. Aside from two structural resonances, an antiresonance defined with equation (A14) is shown.

Another interesting situation occurs when inertance b₁₂ is tuned such that the antiresonance matches one of the two system resonances, see Figure A4.

Figure A4 Shift of the system resonances and induced antiresonance with the increase of inertance b12.

Figure A5 FRF of H2,1 for the case when inertance b12 induces antiresonance that matches (a) the first resonance; (b) the second resonance. Two cases, c1,2 = 0 and c1,2 ≠ 0 are shown.

The two system resonance frequencies can be calculated by equating the denominator of equation (A12) to zero, under the vanishing damping assumption (c_1,2 = 0), which yields:

$${R_1}^2\left(b_{12}\right)=\frac{A+B}{C},$$(A17)

$${R_2}^2\left(b_{12}\right)=\frac{A-B}{C},$$(A18)

where

$$A=\left(k_{12}+k_2\right)m_1+\left(k_1+k_{12}\right)m_2+\left(k_2+k_1\right)b_{12},$$(A19)

$$B={\left({a_1}^2{k_1}^2+k_1\left(2k_{12}\left(a_1{m}_1-a_2{m}_2\right)+2k_2\left(a_1{m}_1+a_2{b}_{12}\right)\right)+{{k_{12}}^2\left(m_1+m_2\right)}^2-2k_{12}{k}_2\left({{b_{12}{m}_2-m}_1}^2-a_1{m}_1\right)+{k_2}^2{\left(m_1+b_{12}\right)}^2\right)}^{\frac{1}{2}},$$(A20)

$$C=2\left({b_{12}m}_2-a_1{m}_1\right),$$(A21)

where a₁ = −m₂ − b₁₂, a₂ = b₁₂ − m₂. In that case, the antiresonance cancels the resonance, effectively producing a 1DOF system. Figure A4 shows cases when the inertance b₁₂ is tuned such that the inerter antiresonance equals to either the first or the second resonance. It can be shown that this condition is satisfied for:

$$b_{12,\mathrm{R}1}=\frac{k_{12}{m}_2}{k_2},$$(A22)

for the first resonance, and

$$b_{12,\mathrm{R}2}=\frac{k_{12}{m}_1}{k_1},$$(A23)

for the second resonance. This is valid if $\sqrt{k_1{m}_1}>\sqrt{k_2{m}_2}$. If $\sqrt{k_1{m}_1}<\sqrt{k_2{m}_2}$, then the inertance values b_12,R1 and b_12,R2 swap. Note however that the two values of the inertance are calculated assuming there is no damping present in the system. As there is always at least a small structural damping in the system, this makes the peak of the cancelled resonance appear in the amplitude and phase plots (see Fig. A3).

Appendix B

The technical drawing of the mechanical part of the experimental test rig is shown in Figure B1. Because the masses and stiffnesses of the leaf springs are significantly smaller than the masses and stiffnesses of two blocks, the first two natural frequencies, as well as their corresponding vibration modes, for the most part agree with ones that would be calculated assuming that the springs do not possess inertia and that the blocks were rigid. However, since the parameters of this system are distributed, additional natural modes appear which cannot be considered using the lumped parameter model at frequencies higher than its second dominant natural frequency (Fig. B2).

Figure B1 Technical drawing of the mechanical part of the experimental setup.

Figure B2 Mode shapes and natural frequencies of the CAD model obtained with FEM. The model is carefully designed to separate the first two main modes from the third and higher modes of vibration.

A convenient technology for the fabrication of such an experimental setup is 3D printing, as it enables a fast transition from a computer-generated model to a physical prototype using a CAD/CAM approach. A fused deposition modelling printer was used and PETG (Polyethylene Terephthalate Glycol modified) filament was used as the material. After the printing was complete, the mechanical part of the prototype was equipped with sensors and actuators. The FEM was used to carefully tune the elements corresponding to mass and stiffness parameters of the system. The damping c_1,2 was determined after conducting the experiments to match the experimental results. The smeared values of material properties (Young modulus E = 1.5 GPa, Poisson coefficient ν = 0.33) of the 3D printed structure were taken from [35].

References

All Tables

All Figures

	Figure 1 The 2DOF mechanical system equipped with a direct acceleration feedback loop, plot (a), with the purpose of synthesizing the inerter, plot (b).
In the text

	Figure 2 Nyquist plot of the open-loop sensor-actuator transfer function assuming the transducers are ideal. The contour never crosses the negative real axis meaning that the system is stable for all gains g of the feedback loop.
In the text

	Figure 3 Closed-loop transfer function for different values of relative acceleration feedback.
In the text

	Figure 4 The Nyquist plot of the sensor actuator open-loop FRF Hs,a obtained theoretically with transducer dynamics included in the feedback loop.
In the text

	Figure 5 The Nyquist plot of the sensor actuator open-loop FRF obtained theoretically (a, c) and experimentally (b, d), with (c, d) and without (a, b) low-pass filter included in the feedback loop. Clockwise rotation of the contour because of the additional low-pass filter is evident. Phase margins are as follows: (a) limit of stability, φ11 = 0; (c) φ21 = 14°; (b) φ12 = 5.95°; (d) φ22 = 10.91°.
In the text

	Figure 6 The transfer accelerance of the active system with increasing feedback gains: (a) up to 10 000 Hz where the accelerometer resonance is visible, (b) up to 800 Hz to capture the main structural modes of the system.
In the text

	Figure 7 The amplitude and phase of the FRF of the active system with low-pass filter included: (a) obtained theoretically, (b) obtained experimentally. The peak at 50 Hz is due to electrical network noise.
In the text

	Figure 8 Amplitude-frequency and phase-frequency characteristics of inertance with included transducer dynamics: (a) amplitude-frequency, (b) phase-frequency.
In the text

	Figure 9 Relation between the modulus of the complex inertance and feedback gain with included transducer dynamics for different frequencies: (a) up to inertance of 0.2 kg, (b) magnified for low values of the feedback gain up to 0.01 kg.
In the text

	Figure 10 The amplitude of the FRF of current and acceleration of the receiving mass with respect to the voltage applied to the primary excitation actuator. The antiresonance frequency for the system at hand is $f_{i_c}={\omega }_{i_c}2\pi =12\pi \sqrt{k_2{m}_2}=61.5 \mathrm{Hz}$.
In the text

	Figure 11 The experimental setup. Accelerometers measuring acceleration signal from both masses is depicted in blue colour, rigid masses and leaf springs are depicted in purple and green colour, respectively. Primary and control actuators are noted in blue. The second actuator is not used in the experimental analysis.
In the text

	Figure 12 Scheme of the experimental setup for measuring the closed-loop response. The acceleration signal subtraction is performed using a simple operational amplifier based signal subtractor.
In the text

	Figure 13 Time domain velocity response of the second mass if the system is turned active in comparison to the passive case. The response is shown for three values of frequencies around the antiresonance frequency: (a) 90 Hz, (b) 85 Hz, (c) 95 Hz.
In the text

	Figure A1 Amplitude of the FRF H2,1 with ideal inerter device, with highlighted high-frequency amplitude.
In the text

	Figure A2 Shift of the high-frequency asymptote with respect to the inertance b12.
In the text

	Figure A3 Amplitude of the FRF $H_{f_c,f_{p1}}$ with ideal inerter device. Aside from two structural resonances, an antiresonance defined with equation (A14) is shown.
In the text

	Figure A4 Shift of the system resonances and induced antiresonance with the increase of inertance b12.
In the text

	Figure A5 FRF of H2,1 for the case when inertance b12 induces antiresonance that matches (a) the first resonance; (b) the second resonance. Two cases, c1,2 = 0 and c1,2 ≠ 0 are shown.
In the text

	Figure B1 Technical drawing of the mechanical part of the experimental setup.
In the text

	Figure B2 Mode shapes and natural frequencies of the CAD model obtained with FEM. The model is carefully designed to separate the first two main modes from the third and higher modes of vibration.
In the text