Introduction

Amplification of the vibrational response of mechanical and structural systems is of key significance to enhance their dynamic properties and upgrade their key performance attributes. To this end, various inertial and displacement amplification mechanisms have been developed and implemented in numerous engineering fields. The performance of such mechanisms has been explored by researchers in various applications including sensors and electromechanical signal amplifiers [1,2,3] energy harvesting mechanisms [4,5,6,7,8,9,10], as well as vibration and sound mitigation technologies [11]. In addition, flexure-based displacement amplifiers, e.g., lever-principle based amplifiers [12,13,14], triangular/arc-principle based amplifiers [15, 16], and compliant mechanisms [17,18,19], have been proposed in the past decades showcasing their significance and cost-effectiveness in various fields such as bioengineering, optical instruments and other ultra-precision and nano-manipulation technologies [20,21,22].

More specifically, vibration control has received considerable research interest in the past few decades, with special emphasis on developing effective, simple, affordable and applicable control systems to mitigate the vibration of mechanical systems, introduce sound attenuation and protect structures against natural or man-made hazards [23,24,25,26]. Vibration control advancements focus lately on the development of passive, semi-active, and active vibration control approaches. Among others, these include the incorporation of additional oscillating masses, that introduce damping to the dynamic system (e.g., Tuned Mass Dampers—TMD) [27,28,29,30], and the application of negative stiffness elements [31,32,33,34,35] (i.e., Negative Stiffness Devices—NS, and Quasi-Zero Stiffness Oscillators—QZS).

Aiming to overcome the hefty mass requirements of these existing systems, various methodologies have been proposed to artificially increase the inertial forces of the oscillating masses and achieve vibration control [12, 36,37,38,39]. Recently, Smith et al. [40] introduced an innovative damping mechanism, namely the inerter, that takes advantage of amplified effective inertia in order to mitigate vibration of various structures. An inerter is a lightweight, linear mechanical device with two terminals that generates a resisting force that is proportional to the relative acceleration of its terminals. Although the system was originally designed for machinery and specifically train and other vehicle suspension mechanisms, its application was extended lately as a solution to numerous vibration control problems.

In particular, such inertial amplification devices have been introduced and experimentally tested as a means to enhance the performance of conventional base isolation and TMD systems [41,42,43,44,45]. As an example, Marian and Giaralis [46] and Giaralis and Taflanidis [47] proposed an inerter-enhanced TMD, namely the tuned-mass-damper inerter, as a seismic protection measure of structures. In a similar way, Cheng et al. [48] presented a simple inertial amplification mechanism, the IAM, that improves the performance of the classic TMD using the amplification effect of a triangular shape mechanical system. In addition, the introduction of negative stiffness (NS) elements has been proposed as an artificial way to increase the inertia of oscillators and to improve the dynamic response of various structures. An example of such an NS-based absorber is the KDamper (KD) concept that has been examined in various engineering applications such as seismic mitigation and protection of structures [49, 50] and low-frequency sound attenuation [51, 52].

The concept of mass amplification has been also considered as a method to enhance the vibration attenuation and filtering properties of phononic crystals and locally resonant metamaterial lattices. A number of studies have revealed the beneficial effects of increasing the inertia of the oscillating masses of a periodic structure with minimal increase of the overall mass of the system [53, 54]. Analysis indicated that large band gaps can be generated and expanded to the low-frequency domain depending on the inerter/amplifier configuration. Yilmaz et al. [55] have analyzed metamaterial lattices with levered amplification mechanisms while Acar and Yilmaz [56] designed a two-dimensional periodic solid structure with embedded inertial amplification mechanisms that was experimentally tested and indicated the existence of large band gaps that extend to the low-frequency domain. In addition, the introduction of negative stiffness elements as a means to optimize the behavior of acoustic metamaterials through phenomenal inertia amplification, has been highlighted in various numerical [57, 58] and experimental studies [59].

In this study, a novel dynamic directional amplifier, namely the DDA mechanism, is introduced aiming to artificially increase the resonating mass of an oscillator, with no requirement of complex geometries and heavy parasitic masses. The mechanism’s rationale lies in the dynamics of a system that is subjected to a holonomic constrain [60, 61]; the vibrating mass is fixed to a simple rigid link that increases inertia towards the desired direction of motion by forcing the oscillator to move through a prescribed circumferential path. The mechanism has been previously introduced by the authors to phononic crystals [62, 63] and locally resonant metamaterial structures [64] to generate bandgaps within the low-frequency domain. It is clearly shown that we can enhance the performance of dynamic structures while retaining mass requirements and complex configurations to a minimum level. The simple geometry and uncomplicated structure of the DDA allows its application to real life structures, as a vibration control mechanism.

Mechanical Design of a DDA Mechanism

The DDA mechanism is a simple structure with main features the mass amplification and the diversion of the output motion of the oscillating mass. The physical model of the DDA is shown in Fig. 1a and comprises the two links AC, A’C’, which are welded to the axle along A-A’. In this way, the links are free to rotate and as consequence the mass attached to the links follows the same rotational movement. The mounting conditions of the bearing shafts are not depicted in this figure; however, it is assumed that they are fixed to a rigid base. A vertical and a horizontal spring provide the necessary stiffness to the system, and the rigid panel (mass) is excited by an applied force (F) towards the x-direction. By all means, other realizations are also feasible; e.g., the entire system could be monolithically manufactured as a compliant structure with flexure hinges [22], however it is deemed at this point that the adopted mechanical design fits better the requirements and purpose of the study.

Fig. 1
figure 1

a 3D realization of the considered Dynamic directional amplification (DDA) mechanism Kinematic model of the Dynamic directional amplification (DDA) mechanism, where the motion \(\mathrm{u}\) of mass \(\mathrm{m}\) is kinematically constrained to the motion \(\mathrm{v}\), b initial mass position c mass position at deformed state

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Theory

Kinematics of the DDA Mechanism

The Cartesian coordinate system is established as shown in Fig. 1. Connecting the mass to the origin of the local coordinate system (CSYS) via a hinged, rigid rod of length AB = L, which is assumed to be massless for the purpose of this analysis, imposes a kinematic constraint between the DoFs u and v, in the x, y direction, respectively. The lumped parameter model is described by the coordinates of the mass (m) at a generic position B(x, y) = (x0 + u, y0 − v), where x0, y0 are the initial coordinates of the mass. The initial angle between the horizontal axis and the link is denoted by φ = arctan(x0/y0), while θ denotes the rotation of the rod at the generic position B of the moving mass. Where kx, ky are the springs stiffnesses, cx, cy the damping coefficient on the horizontal and vertical directions respectively, and F the force exciting the mass in the x direction.

Based on the geometric relationship of the DDA mechanism, the following equation is obtained:

$$x_{0} = \sqrt {L^{2} - y_{0}^{2} }$$
(1)

When the point B moves along x direction with displacement u a coupled motion v along y direction will occur. Hence:

$$x_{0}^{2} + y_{o}^{2} = (x_{0} - u)^{2} + (y_{0} + v)^{2}$$
(2)

To obtain the displacement of point B, Eq. (2) can be simplified and rewritten as

$$u^{2} - 2x_{0} u + 2y_{0} v + v^{2} = 0$$
(3)

The solution of which is

$$v = - y_{0} \pm \sqrt { - u^{2} + 2ux_{0} + y_{0}^{2} }$$
(4)

It can be seen from Eq. (4) that the relationship between the horizontal and vertical displacement (u, v) is nonlinear. For small displacements of the mass m the relationship between u and v (at u = 0) can be expressed after linearization as

$$\frac{u}{v} \cong \frac{{dx_{0} }}{{dy_{0} }} = \partial \left( {\sqrt {L^{2} - y_{0}^{2} } } \right) = - \frac{{y_{0} }}{{\sqrt {L^{2} - y_{0}^{2} } }} = - \frac{{y_{0} }}{{x_{0} }} = - \frac{1}{\tan \varphi }$$
(5)

And for \(\rho = \tan \varphi\) Eq. (5) becomes

$$v = u\rho$$
(6)

The negative sign indicates that a decrease of u results in an increase of v.

Dynamic Modelling

Let T be the kinetic energy of m as follows:

$$T = \frac{1}{2}m(\dot{u}^{2} + \dot{v}^{2} )$$
(7)

Substituting Eq. (5) into Eq. (7), the kinetic energy can be rewritten as

$$T = \frac{1}{2}m\dot{u}^{2} \left( {1 + \tan \varphi } \right)^{2}$$
(8)

Similarly, the potential energy is given as

$$U = \frac{1}{2}k_{{\text{x}}} (L_{{\text{x}}} - L_{{{\text{x}}0}} )^{2} + \frac{1}{2}k_{{\text{y}}} (L_{{\text{y}}} - L_{{{\text{y}}0}} )^{2}$$
(9)

where

$$L_{{\text{x}}} = \sqrt {x^{2} + v^{2} } = \sqrt {(x_{0} - u)^{2} + v^{2} }$$
(10)
$$L_{{\text{y}}} = \sqrt {u^{2} + y^{2} } = \sqrt {(y_{0} + v)^{2} + u^{2} }$$
(11)

Substituting Eqs. (10)–(11) into Eq. (9), and assuming that Lx0 = x0 and Lyo = y0 the potential energy U is

$$U = \frac{1}{2}k_{{\text{x}}} (\sqrt {(x_{0} - u)^{2} + (u\tan \varphi )^{2} } - x_{0} )^{2} + \frac{1}{2}k_{{\text{y}}} (\sqrt {(y_{0} + v)^{2} + u^{2} } - y_{0} )^{2}$$
(12)

The Rayleigh dissipation function of the mechanism is given as

$$D = \frac{1}{2}c_{x} \mathop {\left( {\sqrt {\left( {x_{0} - u} \right)^{2} + \left( {u\tan \varphi } \right)^{2} } - x_{0} } \right)^{2} }\limits^{ \cdot } \, + \,\frac{1}{2}\,c_{y} \mathop {\left( {\sqrt {\left( {y_{0} + v} \right)^{2} + u^{2} } - y_{0} } \right)^{2} }\limits^{ \cdot }$$
(13)

By using of Lagrangian equation,

$$\frac{d}{dt}\left[ {\frac{\partial (T - U)}{{\partial \dot{q}_{i} }}} \right] - \frac{\partial (T - U)}{{\partial q_{i} }} + \frac{\partial D}{{\partial \dot{q}_{i} }} = Q_{i}$$
(14)

The governing equation of the DDA mechanism in the direction of motion is given as follows

$$M\ddot{u} + C\dot{u} + Ku = F$$
(15)

where, \(M = (1 + tan^{2} (\varphi ))m\), \(C = c_{x} + c_{y} tan^{2} (\varphi )\) and \(K = k_{x} + k_{y} tan^{2} (\varphi )\).

And the natural frequency of the mechanism is given as:

$$f_{{{\text{n}},{\text{DDA}}}} = \frac{1}{2\pi }\sqrt {\frac{{(1 + \rho^{2} )m}}{{k_{{\text{x}}} + k_{{\text{y}}} \rho^{2} }}}$$
(16)

It is noted that the angle \(\varphi\) will affect the natural frequency of the DDA mechanism, while for kx = ky the natural frequency of the mechanism is the same with an equivalent SDoF system.

The damping coefficient (C) of the modified system can be calculated as a function of the damping ratio ζn, and can be written as:

$$C = 4\pi mf_{{{\text{n}},{\text{DDA}}}} (\zeta_{{\text{x}}} + \zeta_{{\text{y}}} \rho^{2} )$$
(17)

Transfer Functions

Due to the coupling conditions of u, v; the system can be characterized as an oscillator with one apparent DoF, taking into account Eq. (15). With the application of a Laplace transform the transfer function of the dynamic magnification factor can be expressed as:

$$\tilde{T}_{{{\text{Ux}}}} = \frac{{\tilde{u}}}{{\tilde{u}_{{{\text{ST}}}} }} = \frac{{k_{{\text{x}}} + k_{{\text{y}}} \rho^{2} }}{{ - m\left( {1 + \rho^{2} } \right)\omega^{2} + \left( {c_{{\text{x}}} + c_{{\text{y}}} \rho^{2} } \right)j\omega + \left( {k_{{\text{x}}} + k_{{\text{y}}} \rho^{2} } \right)}}$$
(18)

Another common way to express the amplitude of the response of the system to the amplitude of the excitation is in terms of acceleration to force.

$$\tilde{T}_{{{\text{AFx}}}} = \frac{{\tilde{\ddot{u}}}}{{\tilde{F}}} = \frac{{ - \omega^{2} }}{{ - m\left( {1 + \rho^{2} } \right)\omega^{2} + \left( {c_{{\text{x}}} + c_{{\text{y}}} \rho^{2} } \right)j\omega + \left( {k_{{\text{x}}} + k_{{\text{y}}} \rho^{2} } \right)}}$$
(19)

In the case of base excitation of the system \(a_{B} (t) = \tilde{A}_{B} e^{i\Omega t}\) the transfer functions of the system are

$$\tilde{T}_{{{\text{UBx}}}} = \frac{{\tilde{u}}}{{A_{{\text{B}}} }} = - \frac{{m\left( {1 + \rho^{2} } \right)}}{{ - m\left( {1 + \rho^{2} } \right)\omega^{2} + \left( {c_{{\text{x}}} + c_{{\text{y}}} \rho^{2} } \right)j\omega + \left( {k_{{\text{x}}} + k_{{\text{y}}} \rho^{2} } \right)}}$$
(20)

And

$$\tilde{T}_{{{\text{ABx}}}} = \frac{{\tilde{\ddot{u}}}}{{A_{{\text{B}}} }} = 1 - \omega^{2} \tilde{H}_{{{\text{UBx}}}} = - \frac{{\left( {c_{{\text{x}}} + c_{{\text{y}}} \rho^{2} } \right)j\omega + \left( {k_{{\text{x}}} + k_{{\text{y}}} \rho^{2} } \right)}}{{ - m\left( {1 + \rho^{2} } \right)\omega^{2} + \left( {c_{{\text{x}}} + c_{{\text{y}}} \rho^{2} } \right)j\omega + \left( {k_{{\text{x}}} + k_{{\text{y}}} \rho^{2} } \right)}}$$
(21)

The transfer functions on the y direction can be easily calculated based on Eq. (6) as

$$\left[ {\begin{array}{*{20}c} {\tilde{T}_{{{\text{Uy}}}} } \\ {\tilde{T}_{{{\text{AFy}}}} } \\ {\tilde{T}_{{{\text{UBy}}}} } \\ {\tilde{T}_{{{\text{ABy}}}} } \\ \end{array} } \right] = \rho \left[ {\begin{array}{*{20}c} {\tilde{T}_{{{\text{Ux}}}} } \\ {\tilde{T}_{{{\text{AFx}}}} } \\ {\tilde{T}_{{{\text{UBx}}}} } \\ {\tilde{T}_{{{\text{ABx}}}} } \\ \end{array} } \right]$$
(22)

Parametric Analysis Results

Prior to the main analysis it is worth examining the mass amplification \((1 + \tan^{2} (\varphi ))\) as a function of the amplifier’s angle (φ). For small angles i.e., less than 45°, the mass is amplified less than two (2) times, while for further increase of the angle, the amplification increases exponentially; for \(\varphi\)=75° the mass is amplified 15 times.

Subsequently, numerical simulations were performed to study the response of the DDA mechanism based on the magnification factor (TU) and accelerance (TAF). The analysis parameters are the following: mass equal to m = 0.89 kg, spring stiffness kx = 500N/m and damping ratio ζx = 0.2. These are selected in accordance with the parameters adopted in the experiment, as described in the following sections of this research study.

Figure 2a, b depict the surface plots of accelerance (TAF) and magnification factor (TU), as a function of the spring ratio (ky/kx), assuming that the amplification angle is equal to φ = 75°. As expected, when ky is absent, resonance shifts to lower frequencies while when ky is utilized, the system becomes stiffer and the resonant frequency increases parabolically, as ky is a function of tan2 φ. In Fig. 2c, d the transfer functions are plotted assuming the case that the vertical and horizontal springs have equal stiffness (ky/kx = 1). This allows an illustration of the effect of the amplifier’s angle (φ) compared to the original SDoF oscillator. It is observed that for both transfer functions, the resonant frequency is not affected by the amplifier’s angle and remains the same with the SDoF system.

Fig. 2
figure 2

Surface plot describing a the Frequency response of accelerance (FRF) as a function of the spring’s stiffness ratio ky/kx = 0–2, φ = 75ο and b the magnification factor H as a function of the spring’s stiffness ratio ky/kx = 0–2, φ = 75ο. c Frequency response of accelerance (FRF) and d magnification factor H for stiffness ratio ky/kx = 1 and φ = 30, 45, 75ο compared to the SDoF oscillator. e Frequency response of accelerance (FRF) and f magnification factor H for stiffness ratio ky/kx = 0 and φ = 30, 45, 75ο compared to the SDoF oscillator

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However, as the angle (φ) increases, accelerations are drastically reduced, while displacements follow the same pattern with the SDoF system regardless of the amplifier’s angle (φ). In Fig. 2e, d the spring stiffness ky is set to zero (ky = 0), and consequently, as described in Eq. (16), the resonant frequency of the DDA is determined by the mass amplification \((1 + \rho^{2} )\). Ultimately, the mechanism functions as a classic vibration controller; the increase of the amplifier’s angle leads to a lower resonant frequency which provides isolation both in terms of acceleration and displacement.

Figure 3 illustrates the collateral effect of the out of plane movement of the mechanism. Once again, the transfer functions are presented for two different spring ratios; ky/kx = 1 in Fig. 3a, b, and ky/kx = 0 in Fig. 3c, d. As described in Eq. (22), for large amplifier’s angle (such as φ = 75ο) the out of plane accelerations and displacements are larger than those in the direction of interest. Nonetheless, this should not be considered as a discouraging aspect of the DDA, but rather as an essential attribute that should be carefully examined based on the application of interest and other case dependent restrictions. Specifically, regardless of the spring ratio, the out of plane accelerations are always reduced compared to the ones of the SDoF oscillator, while at the same time the maximum displacements may present an increase, especially for the case of large amplifier’s angles. Obviously, the plot of the SDoF transfer functions along with the out of plane DDA transfer functions should not be considered for direct comparison, as the results refer to different movements, yet this graphical representation allows a better understanding of the overall performance of the mechanism.

Fig. 3
figure 3

a Frequency response of accelerance (FRF) and b magnification factor H for amplifiers angle φ = 75ο and ky/kx = 1. c Frequency response of accelerance (FRF) and d magnification factor H for amplifiers angle φ = 75ο and ky/kx = 0

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Experimental Setup and Testing

Experimental Setup

The performance of the developed Dynamic Directional Amplification mechanism is verified, and the developed theoretical models are validated via experimental testing. The configured experimental setup is presented in Fig. 4. For the experimental measurements, a PCB 356A16 tri-axial accelerometer with 10.2 mV/m/s2 sensitivity is attached to the seismic mass and a PCB 333B30 single-axis accelerometer with 10 mV/m/s2 sensitivity is attached on the shaking table to measure the input motion to the oscillator. The accelerometers and load cells are connected to the computer through two PCB 480B21 amplifiers and run through an NI CB-68LP terminal connector block to the NI PCI-6052E PCI card.

Fig. 4
figure 4

Schematic diagram of the experimental setup for the analysis and validation of the dynamic behavior DDA mechanism

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The recording of the measurements is done via the NI LabView 2013® software, while the post-processing was carried out using in-house developed scripts on the MATLAB R2018a® software.

Figure 5 shows the device with the oscillating mass at the position of equilibrium. The amplifier’s hinges are constructed using bearings and a shaft rod axis. The rigid links are welded to the axis and bolted via an L-shaped section with the mass. Regarding the vertical stiffness elements, two springs are hanged from a stable frame and are attached to the mass through overhangs.

Fig. 5
figure 5

Photograph of the DDA mechanism

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Experimental Results and Discussions

The experimental specimen is fixed on top of a vertical shaking table and a series of sweep tests are conducted by exciting the base of the mechanism. The response of the mass is subsequently evaluated by recording accelerations in both x and y directions via the aforementioned sensors. The adopted oscillating mass is equal to m = 0.89 kg and the stiffness of each one of the two springs is equal to kx = 250 N/m. No extra damping elements have been added; however, a small amount of structural damping is expected to take place due to the material properties and the friction developed within the connections of the device as the mechanism oscillates. Three different amplifier’s angles are tested, namely φ = 30ο, 40ο, 60ο as well as an additional test without the rigid link to model an equivalent SDoF oscillator. The displacement amplitude and the frequency of the excitation are controlled by the built-in control system of the vibration table. For the purposes of the present test, sinusoidal excitations and frequencies of 2–10 Hz are in turn generated by the shaking table. Results are measured in voltage within the time domain and are subsequently converted to acceleration based on the sensitivity of each sensor. Fast Fourier transform (FFT) is deployed to calculate the spectrum at the seismic base (input) and mass (output) of the device.

Figure 6 depicts the Frequency response of accelerance \(|{\widetilde{T}}_{AB}|=|{\widetilde{a}}_{\left(s\right)} /{\widetilde{a}}_{(g)}\)|, where a(s) is the measured acceleration of the mass and a(g) the acceleration of the shaking table (a) at the x- direction and (b) at the y-direction, according to Fig. 1. The continuous lines present the root mean square of amplitudes values of the experimental data while the markers indicate the measured values for each one of the three repetitions that were carried out for each amplifiers angle. The experimental results validate the initial theoretical predictions. The comparison in Fig. 6a with the corresponding Transfer function of the system when the mass is mounted only on the spring elements (SDoF black line) demonstrates the effectiveness of the device. The fundamental resonance peak is reduced, specifically at 3.1 Hz which corresponds to a 19.7% reduction for φ = 30ο and 25.6% for φ = 40ο. By all means, for larger angles the reduction is much greater. Additionally, the extreme peak observed at the SDoF response due to resonance has been diminished and is reduced as the angle (φο) of the DDA increases.

Fig. 6
figure 6

Experimentally measured Transfer functions (ΤΑΒ) of the oscillating mass in a x direction b y direction acceleration to base acceleration of the DDA. The continuous lines show the averaged experimental results and the markers the measured data of each test

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In Fig. 7 the experimentally measured Transfer functions for (a) φ = 30ο and (b) φ = 40ο are provided and compared with the theoretically evaluated response. It is clearly shown that the acceleration transmissibility measured from the experimental process and the one calculated from the analytical solution of the DDA are in overall very good agreement for the prescribed (φ) angles. Specifically, for φ = 30ο the theoretical and experimental response of both the x and y accelerations are almost identical. Hereafter, it is concluded that the analytical equations can be safely used to describe the performance of such an amplification mechanism.

Fig. 7
figure 7

Comparison between averaged experimental results (markers) and theoretically calculated (continuous lines) Transfer functions (ΤΑΒ) for a φ = 30ο b φ = 40ο

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Summary and Conclusions

In conclusion, this study demonstrates a novel dynamic directional amplification mechanism, namely the DDA, that aims to artificially increase the inertia of a simple oscillator and enhance the dynamic properties of vibrating structures. The major innovation of the application lies in the simple geometry of the proposed system; the vibrating mass is fixed to a rigid link that increases inertia towards the desired direction of motion by coupling the kinematic DoFs of the resonating mass and forcing the oscillator to move through a prescribed circumferential path.

An analytical framework has been proposed, providing the theory and mathematical formulation of the DDA mechanism. Transfer functions are derived, and a parametric analysis is subsequently undertaken to evaluate the effect of stiffness and amplifier’s angle to the dynamic response of the oscillating mass. As a following step, the performance of the developed mechanism is verified, and the theoretical models are validated via shaking table experimental testing. A potential disadvantage of the mechanism is the diversion of the mass movement perpendicular to the direction of the applied action; this leads to mass movement in both axes (2D movement in the horizontal and vertical axis) that needs to be accommodated depending on the application and design of the structure.

Specifically, the main outcomes of this work are the following:

  • The introduction of the DDA to the oscillating mass reduces accelerance in the direction of the applied force;

  • The fundamental resonance peak reduces (damping increases) and shifts towards lower frequencies depending on the amplifier’s angle;

  • Results indicate an overall agreement between the experimental and analytically derived calculations.

The proposed framework is presented by the authors as a proof of concept for future studies in applications such as sound and vibration isolators, acoustic panels, acoustic and seismic metamaterials and other vibration control devices that aim to exploit the DDA’s dynamic amplification properties.