Abstract
Purpose
Inertial amplification of an oscillating mass has been considered by various researchers as a means to introduce enhanced vibration control properties to a dynamic system. In this paper an experimental prototype of a novel inertial amplifier, namely the Dynamic Directional Amplification mechanism (DDA), is developed and its dynamic response is subsequently evaluated. The DDA is realized by imposing kinematic constraints to the degrees of freedom (DoFs) of a simple oscillator, hence inertia is increased by coupling the horizontal and vertical motion of the model.
Methods
The concept and mathematical framework of the amplifier are introduced and then validated with experimental measurements conducted on the vertical shaking table, located in the Dynamics & Acoustics Laboratory, National Technical University of Athens.
Results
Analysis indicates the beneficial effect of the DDA to the dynamic response of the oscillator when compared to the initial structure, showcasing a decrease in the acceleration values and shift of the resonating frequency in the derived transfer functions.
Conclusions
The key novelty of the DDA lies in its inertial amplification properties, introduced by a simple geometry and easy-to-apply structure. The proposed framework may be incorporated in applications such as sound and vibration isolators, acoustic panels, acoustic and seismic metamaterials and other vibration control devices that aim to explore the DDA’s dynamic amplification properties. The mechanism has been previously applied by the authors to phononic and locally resonant metamaterials aiming to introduce bandgaps within the low-frequency domain.
Graphical Abstract
![](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs42417-023-00925-5/MediaObjects/42417_2023_925_Figa_HTML.png)
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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.
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
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:
When the point B moves along x direction with displacement u a coupled motion v along y direction will occur. Hence:
To obtain the displacement of point B, Eq. (2) can be simplified and rewritten as
The solution of which is
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
And for \(\rho = \tan \varphi\) Eq. (5) becomes
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:
Substituting Eq. (5) into Eq. (7), the kinetic energy can be rewritten as
Similarly, the potential energy is given as
where
Substituting Eqs. (10)–(11) into Eq. (9), and assuming that Lx0 = x0 and Lyo = y0 the potential energy U is
The Rayleigh dissipation function of the mechanism is given as
By using of Lagrangian equation,
The governing equation of the DDA mechanism in the direction of motion is given as follows
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:
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:
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:
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.
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
And
The transfer functions on the y direction can be easily calculated based on Eq. (6) as
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.
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
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.
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.
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.
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.
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.
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.
Availability of Data and Material
The data used to support the findings of this study are available from the corresponding author upon request.
Code Availability
The code used to support the findings of this study are available from the corresponding author upon request.
References
Iqbal S, Lai YJ, Shakoor RI, Raffi M, Bazaz SA (2021) Design, analysis, and experimental investigation of micro-displacement amplification compliant mechanism for micro-transducers. Rev Sci Instrum. https://doi.org/10.1063/5.0061820
Dolev A, Bucher I (2016) Experimental and numerical validation of digital, electromechanical, parametrically excited amplifiers. J Vib Acoust Trans ASME. https://doi.org/10.1115/1.4033897
Aghamohammadi M, Sorokin V, Mace B (2020) Response of linear parametric amplifiers with arbitrary direct and parametric excitations. Mech Res Commun 109:2–5. https://doi.org/10.1016/j.mechrescom.2020.103585
Adhikari S, Banerjee A (2022) Enhanced low-frequency vibration energy harvesting with inertial amplifiers. J Intell Mater Syst Struct 33:822–838. https://doi.org/10.1177/1045389X211032281
Shahosseini I, Najafi K (2014) Mechanical amplifier for translational kinetic energy harvesters. J Phys Conf Ser. https://doi.org/10.1088/1742-6596/557/1/012135
Yang T, Zhou S, Fang S, Qin W, Inman DJ (2021) Nonlinear vibration energy harvesting and vibration suppression technologies: Designs, analysis, and applications. AIP Publishing LLC. https://doi.org/10.1063/5.0051432
Wang X, Shi Z, Wang J, Xiang H (2016) A stack-based flex-compressive piezoelectric energy harvesting cell for large quasi-static loads. Smart Mater Struct. https://doi.org/10.1088/0964-1726/25/5/055005
Miranda R, Babilio E, Singh N, Santos F, Fraternali F (2020) Mechanics of smart origami sunscreens with energy harvesting ability. Mech Res Commun. https://doi.org/10.1016/j.mechrescom.2020.103503
Dai W, Yang J (2021) Vibration transmission and energy flow of impact oscillators with nonlinear motion constraints created by diamond-shaped linkage mechanism. Int J Mech Sci 194:106212. https://doi.org/10.1016/j.ijmecsci.2020.106212
Ha L, Fang LJ (2014) Error analysis of a non-contact parallel plane sensor based on Monte Carlo method. Adv Mater Res 1022:96–99. https://doi.org/10.4028/www.scientific.net/AMR.1022.96
Bergamini A, Miniaci M, Delpero T, Tallarico D, Van Damme B, Hannema G, Leibacher I, Zemp A (2019) Tacticity in chiral phononic crystals. Nat Commun 10:4525. https://doi.org/10.1038/s41467-019-12587-7
Flannelly WG (1967) Dynamic antiresonant vibration isolator, U.S. Patent No. 3,322,379
Yilmaz C, Kikuchi N (2006) Analysis and design of passive band-stop filter-type vibration isolators for low-frequency applications. J Sound Vib 291:1004–1028. https://doi.org/10.1016/j.jsv.2005.07.019
Jones R, Mcgarvey JH (1976) Helicopter rotor isolation evaluation utilizing the dynamic antiresonant vibration isolator. SAE Tech Pap. https://doi.org/10.4271/760894
Li H, Guo F, Wang Y, Wang Z, Li C, Ling M, Hao G (2022) Design and modeling of a compact compliant stroke amplification mechanism with completely distributed compliance for ground-mounted actuators. Mech Mach Theory 167:104566. https://doi.org/10.1016/J.MECHMACHTHEORY.2021.104566
Chen G, Ma Y, Li J (2016) A tensural displacement amplifier employing elliptic-arc flexure hinges. Sens Actuators A Phys 247:307–315. https://doi.org/10.1016/J.SNA.2016.05.015
Guo F, Sun Z, Zhang S, Cao R, Li H (2022) Optimal design and reliability analysis of a compliant stroke amplification mechanism. Mech Mach Theory 171:104748. https://doi.org/10.1016/J.MECHMACHTHEORY.2022.104748
Zhu W-L, Zhu Z, Shi Y, Wang X, Guan K, Ju B-F (2016) Design, modeling, analysis and testing of a novel piezo-actuated XY compliant mechanism for large workspace nano-positioning. Smart Mater Struct. https://doi.org/10.1088/0964-1726/25/11/115033
Sun X, Wang J, Yi S, Hu W (2022) Design and analysis of a novel piezoelectric inertial actuator with large stepping displacement amplified by compliant mechanism. Microsyst Technol 28:1025–1035. https://doi.org/10.1007/s00542-022-05257-0
Chen F, Zhang Q, Gao Y, Dong W (2020) A review on the flexure-based displacement amplification mechanisms. IEEE Access 8:205919–205937. https://doi.org/10.1109/ACCESS.2020.3037827
Hong Y, Wu Y, Jin S, Liu D, Chi B (2022) Design and Analysis of a Microgripper with Three-Stage Amplification Mechanism for Micromanipulation. Micromachines. https://doi.org/10.3390/mi13030366
Chen C-M, Hsu Y-C, Fung R-F (2012) System identification of a Scott-Russell amplifying mechanism with offset driven by a piezoelectric actuator. Appl Math Model 36:2788–2802. https://doi.org/10.1016/j.apm.2011.09.064
Ma R, Bi K, Hao H (2021) Inerter-based structural vibration control: a state-of-the-art review. Eng Struct 243:112655. https://doi.org/10.1016/j.engstruct.2021.112655
Elias S, Matsagar V (2017) Research developments in vibration control of structures using passive tuned mass dampers. Annu Rev Control 44:129–156. https://doi.org/10.1016/j.arcontrol.2017.09.015
Balaji PS, Karthik SelvaKumar K (2021) Applications of nonlinearity in passive vibration control: a review. J Vib Eng Technol 9:183–213 https://doi.org/10.1007/s42417-020-00216-3
Lee CM, Goverdovskiy VN, Sotenko AV (2016) Helicopter vibration isolation: design approach and test results. J Sound Vib 366:15–26. https://doi.org/10.1016/J.JSV.2015.08.024
Den Hartog JP (1956) Mechanical vibrations, 4th edn. McGraw-Hill, New York. https://doi.org/10.1038/161503c0
Frahm H (1911) Device for damping of bodies, US patent #989958
Qin L, Yan W, Li Y (2009) Design of frictional pendulum TMD and its wind control effectiveness. J Earthq Eng Eng Vib 29:153–157
McNamara RJ (1977) Tuned mass dampers for buildings. J Struct Div 103:1785–1798
Molyneaux W (1957) Supports for vibration isolation. ARC/CP-322, Aer Res Council, G Britain
Pasala DT, Sarlis A, Nagarajaiah S, Reinhorn A, Constantinou M, Taylor D (2013) Adaptive negative stiffness: new structural modification approach for seismic protection. Adv Mater Res. https://doi.org/10.4028/www.scientific.net/AMR.639-640.54
Sarlis AA, Pasala DTR, Constantinou MC, Reinhorn AM, Nagarajaiah S, Taylor DP (2013) Negative stiffness device for seismic protection of structures. J Struct Eng 139:1124–1133
Ye K, Ji JC, Brown T (2020) Design of a quasi-zero stiffness isolation system for supporting different loads. J Sound Vib 471:115198. https://doi.org/10.1016/J.JSV.2020.115198
Liu X, Huang X, Hua H (2013) On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. J Sound Vib 332:3359–3376. https://doi.org/10.1016/j.jsv.2012.10.037
Desjardins RA (1977) Vibration isolation system, U.S. Patent No. 4,140,028
Desjardins RA, Hooper WE (1980) Antiresonant rotor isolation for vibration reduction. J Am Helicopter Soc 25:46–55. https://doi.org/10.4050/JAHS.25.46
Ivovich VA, Savovich MK (2015) Isolation of floor machines by lever-type inertial vibration corrector. Proc Inst Civil Eng Struct Build 146:391–402. https://doi.org/10.1680/STBU.2001.146.4.391
Braun D (1982) Development of antiresonance force isolators for hellicopter vibration reduction. J Am Helicopter Soc. https://doi.org/10.4050/JAHS.27.37
Smith MC (2002) Synthesis of mechanical networks: the inerter. IEEE Trans Automat Contr 47:1648–1662. https://doi.org/10.1109/TAC.2002.803532
Chowdhury S, Banerjee A, Adhikari S (2021) Enhanced seismic base isolation using inertial amplifiers. Structures 33:1340–1353. https://doi.org/10.1016/j.istruc.2021.04.089
Shi B, Dai W, Yang J (2022) Performance analysis of a nonlinear inerter-based vibration isolator with inerter embedded in a linkage mechanism. Nonlinear Dyn. https://doi.org/10.1007/s11071-022-07564-7
Moraes FH, Silveira M, Gonçalves PJP (2018) On the dynamics of a vibration isolator with geometrically nonlinear inerter. Nonlinear Dyn 93:1325–1340. https://doi.org/10.1007/s11071-018-4262-6
Nakamura Y, Fukukita A, Tamura K, Yamazaki I, Matsuoka T, Hiramoto K, Sunakoda K (2014) Seismic response control using electromagnetic inertial mass dampers. Earthq Eng Struct Dyn 43:507–527. https://doi.org/10.1002/eqe.2355
Jangid RS (2021) Optimum tuned inerter damper for base-isolated structures. J Vib Eng Technol 9:1483–1497. https://doi.org/10.1007/s42417-021-00309-7
Marian L, Giaralis A (2014) Optimal design of a novel tuned mass-damper-inerter (TMDI) passive vibration control configuration for stochastically support-excited structural systems. Probabilistic Eng Mech 38:156–164. https://doi.org/10.1016/j.probengmech.2014.03.007
Giaralis A, Taflanidis AA (2018) Optimal tuned mass-damper-inerter (TMDI) design for seismically excited MDOF structures with model uncertainties based on reliability criteria. Struct Control Heal Monit 25:1–22. https://doi.org/10.1002/stc.2082
Cheng Z, Palermo A, Shi Z, Marzani A (2020) Enhanced tuned mass damper using an inertial amplification mechanism. J Sound Vib 475:115267
Kapasakalis KA, Antoniadis IA, Sapountzakis EJ (2021) Feasibility assessment of stiff seismic base absorbers. J Vib Eng Technol 2021:1–17. https://doi.org/10.1007/S42417-021-00362-2
Mantakas AG, Kapasakalis KA, Alvertos AE, Antoniadis IA, Sapountzakis EJ (2022) A negative stiffness dynamic base absorber for seismic retrofitting of residential buildings. Struct Control Heal Monit. 29:e3127. https://doi.org/10.1002/STC.3127
Paradeisiotis A, Kalderon M, Antoniadis I, Fouriki L (2020) Acoustic Performance Evaluation of a panel utilizing negative stifffness mounting for low frequency noise control. In: Proceedings EURODYN 2020, EASD Procedia, Athens, Greece, 23–26 November, pp 4093–4110
Kalderon M, Paradeisiotis A, Antoniadis I (2021) A Meta-structure for Low-frequency Acoustic Treatment Based on a KDamper-Inertial Amplification Concept. In: Euronoise, pp 1333–1343
Shoaib M, Chen Z, Li F (2021) Vibration attenuation of periodic non-uniform pipes conveying fluid. J Vib Eng Technol 9:2035–2045. https://doi.org/10.1007/s42417-021-00347-1
Frandsen NMM, Bilal OR, Jensen JS, Hussein MI (2016) Inertial amplification of continuous structures: Large band gaps from small masses. J Appl Phys 119:124902
Yilmaz C, Hulbert GM, Kikuchi N (2007) Phononic band gaps induced by inertial amplification in periodic media. Phys Rev B 76:54309. https://doi.org/10.1103/PhysRevB.76.054309
Acar G, Yilmaz C (2013) Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in two-dimensional solid structures. J Sound Vib 332:6389–6404
Paradeisiotis A, Kalderon M, Antoniadis I (2021) Advanced negative stiffness absorber for low-frequency noise insulation of panels. AIP Adv 11:65003. https://doi.org/10.1063/5.0045937
Antoniadis I, Paradeisiotis A (2018) A periodic acoustic meta-material concept incorporating negative stiffness elements for low-frequency acoustic insulation/absorption. In: Proc. ISMA 2018 - Int. Conf. Noise Vib. Eng. USD 2018 - Int. Conf. Uncertain. Struct. Dyn, pp 1179–1193
Chondrogiannis KA, Colombi A, Dertimanis V, Chatzi E (2022) Computational verification and experimental validation of the vibration-attenuation properties of a geometrically nonlinear metamaterial design. Phys Rev Appl 17:54023. https://doi.org/10.1103/PhysRevApplied.17.054023
Udwadia FE, Kalaba RE (1995) An alternate proof for the equation of motion for constrained mechanical systems. Appl Math Comput 70:339–342. https://doi.org/10.1016/0096-3003(94)00113-I
Udwadia FE, Kalaba RE (1992) A new perspective on constrained motion. Proc R Soc Lond Ser A Math Phys Sci. 439:407–410. https://doi.org/10.1098/rspa.1992.0158
Kalderon M, Paradeisiotis A, Antoniadis I (2021) 2D dynamic directional amplification (DDA) in phononic metamaterials. Materials (Basel). https://doi.org/10.3390/ma14092302
Kalderon M, Paradeisiotis A, Antoniadis IA (2022) A Phononic Metamaterial Incorporating Directional Amplification for Low Frequency Isolation. In: Proc. Int. Conf. Nat. Hazards Infrastruct
Kalderon M, Mantakas A, Paradeisiotis A, Antoniadis I, Sapountzakis EJ (2022) Locally resonant metamaterials utilizing dynamic directional amplification: an application for seismic mitigation. Appl Math Model 110:1–16. https://doi.org/10.1016/J.APM.2022.05.037
Acknowledgements
The authors would like to thank Mr. A. Petas for his contribution in the construction of the DDA experimental setup.
Funding
Open access funding provided by HEAL-Link Greece. Moris Kalderon and Antonios Mantakas would like to acknowledge the financial support provided by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant (Grant Agreement No. INSPIRE-813424, “INSPIRE—Innovative Ground Interface Concepts for Structure Protection”).
Author information
Authors and Affiliations
Contributions
MK: Conceptualization, Methodology, Software, Formal analysis, Investigation, Resources, Data curation, Writing—original draft, Visualization. AM: Methodology, Formal analysis, Investigation, Resources, Data curation, Writing—original draft, Visualization. IA: Supervision, Project administration, Funding acquisition.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare there are no conflicts of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kalderon, M., Mantakas, A. & Antoniadis, I. Dynamic Modelling and Experimental Testing of a Dynamic Directional Amplification Mechanism for Vibration Mitigation. J. Vib. Eng. Technol. 12, 1551–1562 (2024). https://doi.org/10.1007/s42417-023-00925-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42417-023-00925-5