Advanced negative stiffness absorber for low-frequency noise insulation of panels

Designing lightweight structures with low vibration and sound radiation has always been technically demanding and practically important. Hence, the need for low-cost and low-mass vibration isolation within the modern aerospace, automotive, and building industries has recently motivated research groups at a worldwide level to develop a number of novel vibration isolation concepts. However, investigation of the acoustic performance of common means of acoustic treatment, such as plasterboard panels, is most often focused in middle to high frequency regions and, particularly, in the coincidence region of sound transmission. Meanwhile, the sound transmission in the critical frequency range of 0–200 Hz has only recently begun to be addressed frequently. Generally, since the “mass law” that governs the mid-frequency range above the fundamental frequency states that the level of sound transmission loss (STL) can be improved only by increasing the mass density of the panel, this leaves only the mitigation of the fundamental panel resonance region, as in Fig. 1, especially when the addition of extra mass needs to be avoided.

Sandwich panels¹ are considered the most common solution, yet their effectiveness in low-frequency isolation has not been extensively documented, even in cases where novel composite honeycomb arrangements² with negative stiffness element inclusions³ are considered. On the other hand, acoustic meta-materials⁴ lead the way toward feasible designs that can be implemented in a wide range of applications. Structural designs based on the so-called acoustic black hole (ABH) phenomenon^5–7 offer remarkable possibilities to manipulate bending waves inside lightweight structures. However, the effective frequency range of conventional ABH structures is limited by the characteristic cut-on frequency, hampering practical applications. Membrane-type acoustic meta-materials,^8–12 usually comprising of resonating masses, have also indicated encouraging results, mainly due to their slender dimensions and isolation properties in low frequencies. Nonetheless, conventional locally resonant meta-materials usually require relatively heavy, additional, internally moving masses, as well as amplitude constraints of these locally resonating structures, which may prohibit their practical implementation. Indicatively, current applications of locally resonant meta-materials in acoustics^13,14 often address frequencies well above 500 Hz.

The application of the KDamper (KD) concept toward the design of highly dissipative low-frequency elastic/acoustic meta-materials^15,16 shows promise in addressing this issue to a certain extent. The KDamper^17,18 is a novel passive vibration isolation and damping concept based essentially on the optimal combination of appropriate stiffness elements, which includes a negative stiffness element. The initial form of the KDamper was optimized to minimize the displacement of the seismic mass under harmonic excitation and was geared more toward applications for machine mounting.

Acknowledging the manufacturing difficulties of meta-materials, especially in conventional structures where the cost is the main limitation, the KD concept has also been investigated for the design of mounting systems suitable for conventional panels.^19,20 These initial investigations indicated the potential of the proposed system in effectively mitigating sound transmission at the fundamental resonance region for target frequencies below 100 Hz.

The concept of inertial amplification goes back many decades. An initial design was based on a vibration isolator consisting of a levered mass in parallel with a spring²¹ amplifying the motion of the added mass by generating large inertial forces and increasing the overall inertia of the system. Inspired by the successful application of inerters in engineering applications, several works expanded the similar concept of inertial amplification mechanisms (IAMs) for formation of finite periodic structures and bandgap generation. The fundamental theory regarding the generation of phononic gaps utilizing IAMs²² was later tested by a first design with flexural hinges and experimentally validated.²³ Further developments included the experimental validation of a 2D periodic structure using shape-optimized IAMs with flexural hinges,²⁴ obtaining wide and deep bandgaps at low frequencies. Recently, a coupled IAM-Tuned Mass Damper (TMD) system²⁵ was proposed for earthquake engineering applications.

The effectiveness and characteristics of the IAM concept can also be utilized in low-frequency noise insulation. As proposed in this work, in combination with the properties of the KDamper, the width and depth of the generated low-frequency attenuation band can be further improved. An extended KDamper framework that incorporates an IAM is thus presented. This coupled absorber is applied in a novel mounting system for flexible acoustic panels, resulting in a wide and deep frequency band of improved vibration and noise attenuation.

The methodology employed in this work is based on a lumped parameter model (LPM) utilizing a first mode approximation for the STL of a simply supported panel, and an optimization procedure is formulated for the selection of the KD-IAM parameters. One of the new developments presented in this work is the refinement of the initial optimization procedure of the KDamper,²⁰ aiming at minimizing the STL in the vicinity of the fundamental resonance. The theoretical framework has been expanded for improved modeling of the fluid–structure interaction of the panel and subsequent representation in terms of a dynamic system with discrete degrees of freedom (DoFs).

Although extensive work has been done in modeling and evaluating the acoustic behavior of structural elements in the low-frequency range, often the methods employed are either complicated theoretical procedures^26,27 or straightforward finite element (FE) analyses, especially regarding the effects of boundary conditions of such structural elements. One of the goals of this work is to highlight as rudimentarily as possible the physical mechanisms affecting the acoustic performance of radiating surfaces in the low- to mid-frequency regions and mainly around the fundamental resonance region. Furthermore, the LPM is a convenient and fast approximative method to evaluate the STL in this region based on a couple of transfer functions, especially in cases of complex boundary conditions and/or periodic structures with multiple degrees of freedom, where distinction needs to be made between localized masses and masses corresponding to radiating surfaces.

The Free Finite Rigid (FFR) panel approximation presented in Subsection II C is essentially a modification of the mass density law, extended to cover also the fundamental resonance region in order to note its detrimental effect on STL. Additionally, it is used to highlight the limitation of the conventional acoustic treatment, in the way that it acts as an upper limit of the achievable STL. This approximation in combination with the LPM aims to show how this limit can only be surpassed with some kind of “meta-structure” exhibiting properties that conventional materials do not possess. Hence, this is the concept proposed in this manuscript.

Investigation also shows that increasing the rigidity of the panel improves the attenuation band. Based on this consequence, an indicative implementation for an initial case study of a stiffened plasterboard panel is demonstrated together with an indicative design and realization of the KD-IAM mounts.

This case study of a simplified, more stripped down application aims to demonstrate the capability of this advanced negative stiffness absorber in the area of low-frequency noise mitigation. Additionally, the basic analytical framework laid here is intended to serve as a basis for future more detailed investigations and designs verified by Finite Element Analysis (FEA) and experimental testing.

Section V contains a brief presentation of a possible meta-structure based on the periodic repetition of KD-IAM unit cells. The results show further improvements, mainly regarding the damping capabilities of low vibrational modes for increasing number of unit cells, indicating the potential of its utilization in the formation of acoustic meta-materials.

By defining the transmission coefficient as

where Π_i is the incident and Π_t is the transmitted (radiated) sound power, the STL is expressed as

Figure 1 shows a typical form of the STL curve for sound propagation through a thin panel.²⁸ In the “mass law” region, the STL is controlled by the mass per unit area of the panel. Specifically, for a thin panel, neglecting stiffness and damping (limp wall),

where Ω is the angular frequency of the propagated incident sound waves, θ_i is the angle of incidence, c₀ is the speed of sound in air, ρ₀ is the air density, and $m̄$ (kg m⁻²) is the mass per unit area of the panel. Although the mass law is very simple and compact, it is only valid for large simple panels because it is derived from the infinite and rigid panel assumption. Therefore, attention should be paid when utilized for STL predictions of general finite panels.

The fluid domain is assumed homogeneous and compressible, while an infinitely large acoustic rigid baffle divides the space into two domains. The mass density of air and sound speed are denoted as ρ₀ and c₀, respectively. The panel is subjected to incident pressure p_i at incidence angle θ_i and azimuth angle ϕ_i. Therefore, the excitation domain consists in the incident pressure p_i, the reflected pressure p_r, and the re-radiated pressure $prad−$ due to the motion of the finite panel. The receiver domain consists in only the transmitted pressure p_t, namely, the radiated pressure due to the motion of the panel $prad+$⁠. Assuming that the incident sound pressure p_i is a harmonic function, we can express it as

where $p̃i(x,y,z)$ is the time invariable complex amplitude and κ = Ω/c₀ is the wavenumber with

The transmitted pressure $ptr,t$ is calculated using Rayleigh’s integral²⁹ in terms of the structural velocity of the panel. Applying the far-field approximation, the transmitted intensity I_t is calculated as

Then, the transmitted power Π_t is determined by integrating the transmitted acoustic intensity on a hemisphere in the far field enclosing the panel, assuming a weak coupling between the vibrating structure and the radiated sound field,

while the incident power on the panel is defined as

With reference to Fig. 2, the finite-sized flat panel partition is assumed to be rectangular and baffled, with lengths l_x and l_y along the x and y axes, respectively. The panel of thickness h is considered homogeneous and isotropic and is modeled as a classical thin plate, implying that the effects of both the rotary inertia and transverse shear deformation can be neglected. The equation of motion governing the bending vibration of the plate is given by^29,30

where $D=Ẽh3/12(1−ν2)$ is the bending stiffness of the plate and E, h, ρ, and ν are the Young’s modulus, thickness, mass density, and Poisson’s ratio of the plate, respectively, while w(ξ, η, t) is the instantaneous transverse displacement. In order to account for energy dissipation due to structural damping, a complex modulus of elasticity is introduced, $Ẽ=E(1+jn)$⁠, where n is the loss factor.

The so-called “blocked pressure”³⁰ p_b = 2p_i is also defined, which is the pressure when the incident wave meets a rigid wall. For simplification, the term 2p_t(ξ, η, t) on the right-hand side can be neglected without significant loss of accuracy for the purpose of this analysis.³¹ Therefore, the blocked pressure p_b is considered as the exciting pressure in the following models.

Considering simply supported boundary conditions along the peripheral edges of the panel, the natural frequency of the first mode results from the homogeneous form of Eq. (9),

Analytic solutions for the STL can be found via modal superposition methods, for example, that formulated by Roussos.³¹ However, since the low-frequency range of interest in the present examination is the range near the fundamental resonance of the panel, an approximation is formulated in this section, considering only the first vibrational mode of the simply supported panel.

Expressing the dynamic behavior of the continuous system of the simply supported panel approximated by the first mode, as a single degree of freedom (SDoF) lumped parameter model (LPM) and assuming normal incident waves (θ_i = 0), the transmission coefficient τ becomes

where ω₁ is the natural frequency of the first mode and m₁ is the generalized mass calculated as

The term I_θϕ is the double integral over a hemispheric surface in the receiver domain, given by

where I_ξη is a surface integral calculated as

where σ_x and σ_y are

The objective of the simplified models is to provide a more straightforward and fast prediction of the acoustic performance of the panel along with the ability to extract intuitive and revealing expressions of the main dynamic parameters that govern its frequency response. With that in mind, the approximation using only the first mode is considered appropriate in order to study this low-frequency region and, subsequently, the optimization and effect of the elastic mounting.

In order to replace the infinite panel approximation (limp wall) with a more appropriate reference curve for finite panels, the free finite rigid (FFR) panel approximation is formulated. In this case, it is assumed that the panel is rigid, consisting only of mass and oscillating freely in an infinite rigid baffle but instead with finite dimensions.

Since the panel is considered rigid, the spatial derivatives of Eq. (9) are eliminated. Considering only the blocked pressure p_b = 2p_i as the forcing pressure and that p_i are normal incident plane waves, the resulting transmission coefficient is

where $m̄=ρh$ is the mass density (mass per surface area) of the panel and I_θϕ is defined as in Eq. (13), where the surface integral in this case is

where σ_x and σ_y are the same as in Eq. (15).

The mathematical model described in this section is an extension of previous work.^19,32 The exact theoretical calculation of how the bending stiffness of the panel combines with the stiffness of the supporting elastic mounts is a cumbersome and complicated procedure.^26,27 The goal here is to formulate a straightforward method to approximate this coupling. This facilitates the following optimization procedure and the investigation of the resulting acoustic performance. In order to do that, the main assumption is that the bending stiffness of the deformable plate $k1=ω12m1$ is in a way in series with the stiffness of the mounting. Therefore, the resulting stiffness is

When the mounting is very stiff (k₀ ≫ k₁), it leads to k_tot = k₁, namely, the case of the simply supported plate is approached. When k₀ → 0, the plate is essentially free floating. In the case that the mounting stiffness k₀ is of comparative order of magnitude with k₁, there is some deviation between the resulting eigenfrequency of the model and the actual eigenfrequency. However, in cases when k₀ ≫ k₁ or k₀ ≪ k₁, the deviation is nullified.

The inclusion of stiffeners as in Sec. IV A for increasing the panel’s rigidity, and hence its fundamental eigenfrequency, helps reinforce this “in-series” approximation. Namely, with the lumped parameter models, it is essentially assumed that the panel is rigid in terms of that the response is uniform along the surface of the panel. Therefore, increasing the rigidity is an attempt to approximate the rigid assumption regarding the displacement/velocity amplitude distribution. Thus, the model can be used for estimation of the required stiffness level and guidance in designing the mounts.

Figure 3(a) shows the equivalent dynamic SDOF model of the deformable thin panel when it is supported on elastic mounts according to the “in-series” assumption. This is achieved by utilizing the generalized values of the structure, namely, the generalized mass and the corresponding generalized stiffness for the first mode of the simply supported panel.

Following the logic of classic vibration isolation, the fundamental eigenfrequency of the panel may be reduced utilizing elastic mounts so that it can be moved outside the frequency range of interest, e.g., above 20 Hz. However, significantly decreasing this frequency can have certain implications. On one hand, it leads to slightly reduced STL frequency response, and on the other hand, it may present practical problems including excessive vibration magnitude; possible resonances with other supportive structural elements, such as bolts or the frame; or even inadequate rigidity for impact loads.

Additionally, the positions of the mounts have a significant role in their damping effectiveness, especially considering that the displacement and velocity amplitudes are maximum at the center of the panel and minimum along the edges. Therefore, a mount positioned at the center would most effectively dampen the fundamental resonance. However, this case would change the dynamic behavior of the panel and would also require an additional supporting frame along the middle of the panel, for example, which is something that is preferably avoided.

The KDamper vibration absorption and damping concept^17,18 is proposed as an alternative realization of elastic mounts. The KDamper is essentially an extension of the traditional Tuned Mass Damper (TMD), with the inclusion of a negative stiffness element. This way, the inertial force of the additional mass of the TMD is supplemented by the force of this negative stiffness element that connects the additional mass with the ground. The overall static stiffness of the oscillator is maintained while the individual stiffness elements are optimally selected. This is important, particularly, when considering the static load bearing capacity of the mount. The KDamper exhibits significantly higher modal damping than the TMD and achieves greater attenuation in a wider frequency band while utilizing significantly lower additional mass.

The static stiffness k₀ of the KDamper is defined as

where f₀ can be seen as a central or design frequency of the KDamper. Normally, in previous vibration absorption applications of the KDamper,^17–20,33f₀ corresponded to the operational excitation frequency of the system in question, around which the attenuation band would be centered. However, in this investigation, f₀ is set as one of the optimization variables for maximization of the STL of the mounted panel.

The stiffness parameters of the KDamper are selected according to Eq. (20) in order to maintain the total stiffness of the system. However, an increase in the absolute value of k_N, or reduction in the absolute value of k_P or k_S, may endanger the static stability of the system. These parameters may present fluctuations due to material fatigue, manufacturing tolerances, non-linear behavior of structural elements, and temperature variations, among others.

In order to ensure that the potential loss of the static stability is prevented, the possible variations of k_N, k_P, and k_S are taken into account during the optimization procedure of the KDamper parameters,³⁴ contrary to the classic KDamper design, which foresees variation only in the negative stiffness element k_N. In that, these static stability margins are imposed via the introduction of the tolerances ϵ_N, ϵ_P, and ϵ_S, respectively, as

so that the limiting values of the stiffness elements are expressed as

Defining the parameters

the neutral point for static stability corresponds to

Consequently, for an assumed value of the non-dimensional parameter κ_N = k_N/k₀—or the absolute value of the negative stiffness element k_N—and selected tolerances, the rest of the stiffness elements result as follows:

where κ_S is the non-dimensional parameter of the k_S stiffness element and

while the non-dimensional κ_P is

Then, the stiffness elements are calculated as

The non-dimensional parameter ρ of the KDamper is also defined as

where m_D = μ m₁. The parameter ρ essentially represents the “distance” between the two eigenfrequencies of the KDamper, which is a marker of the absorption frequency band of the oscillator.

Since the loss factor (n) can represent more accurately the dynamic response of nonlinear systems compared with the damping ratio, which is defined on the grounds of the linear single degree of freedom (SDOF) viscous model,^35,36 hysteretic damping is introduced indirectly considering complex stiffness elements as

The KDamper theoretical framework^17–19 is extended to include the case where the rigid mass is replaced by a deformable plate. Figure 3(b) shows the corresponding model when the equivalent elastic mount is realized based on the KDamper with the inclusion of the inertial amplifier (IAM), where m_j is the connecting mass, which is assumed to be negligible in order to act just as a connector, namely, an additional DOF. Figure 3(c) shows an equivalent model to Fig. 3(b), considering that the effective mass of the IAM configuration is superimposed to the connecting mass and, in total, denoted as m_B.

The aim is the selection of optimal values for the KDamper and IAM elements in order to obtain the maximum possible STL performance, especially in the resonance region of the system in comparison to the initial case where the plate is simply supported on its edges.

The effective mass of the IAM is calculated as²⁵ 

The equations of motion resulting from the model of the panel mounted on KDamper mounts, as illustrated in Fig. 3(c), are

where the generalized degrees of freedom are defined as

where C_i is the corresponding amplitude of motion.

The relevant transfer functions of the model in Fig. 3(c) are

The STL of the KDamper model is then calculated as $STLKD=10log101τKD$⁠, where the transmission coefficient τ_KD is given as

For optimization, the “fmincon” function of the MATLAB software for local minimization of non-linear functions and constraints is utilized. The vector x contains the optimization variables, and f(x) is the objective function to be minimized. In this case, the vector x is

The minimum value of the STL of the model in Fig. 3(b) is considered as the objective function, the algorithm of which is set to maximize, namely,

Therefore, the optimization problem becomes

with the optimization variables between the lower bounds lb = {−0.8 10 0}^T and the upper bounds ub = {0 120 10}^T.

Some of the choices regarding the constraints and the bounds of the optimization variables result from past investigations of the KDamper and relate closely to considerations regarding possible practical implementations,^15,19 which are beyond the scope of the present contribution.

In this section, an examination of the KD-IAM concept is presented, considering the low-frequency acoustic performance of a stiffened panel. During this approach toward increasing low-frequency noise insulation, instead of lowering the fundamental resonance of the structure and taking advantage of the STL region governed by the mass law, the aim is to lift the panel’s fundamental resonance and take advantage of the absorption properties of the mounts in the resonance region. A common practice to enhance the rigidity of a panel involves incorporating a grid of stiffeners, where the density and shape of these ribs define the enhanced stiffness of the system.

The panel under investigation is assumed to comprise a conventional plasterboard enhanced by stiffeners to improve the panel’s rigidity and manipulate the value of its fundamental eigenfrequency. The relevant properties of the plasterboard are summarized in Table I.

In order to investigate the effect of the panel rigidity on the STL frequency response, the KD-IAM model is employed assuming four different arrangements. The first case corresponds to a non-enhanced plasterboard with a fundamental eigenfrequency of 10.8 Hz, while the next three cases correspond to stiffened panels with fundamental eigenfrequencies of 30, 50, and 100 Hz, assuming initially simply supported boundary conditions for the panel and consequent calculation of its generalized values to be used in the optimization procedure.

The red dashed line represents the FFR panel approximation as given by Eq. (17). Intuitively, this approximation should constitute the ideal case regarding the STL of a panel with finite dimensions and may act as a reference curve for various comparisons instead of the infinite panel approximation, which is invalid in the lower frequency range that is being examined. The analysis is conducted for both KDamper mounts without the inclusion of the IAM (m_B = 0) [see Fig. 4(a)] and KD-IAM mounts [see Fig. 4(b)] mounts in order to highlight additional effects. The results for each case obtained after the optimization procedure are given in Tables II and III.

The results of Fig. 4(a) indicate that the KD mounted panel is effectively dampened in the resonance region so that the STL level is improved compared to the FFR panel. As expected, all curves coincide for higher frequencies as the mass law prevails. The capabilities of the KD mounts are displayed, especially in the case of f₁ = 100 Hz where the resonance effect on the STL is almost eliminated and, at the same time, an increased attenuation band is formed in the region. Resonances of higher modes that would appear while using a more extensive modeling approach are also mitigated without a significant amount of damping, especially compared to the fundamental resonance.

Considering the case where the mounts are positioned along the edges of the real panel, the displacement amplitude at the edges is lower than the mean amplitude of the panel due to the shape function of the first mode. Thus, the damping will be less effective in reality. For this reason, the considered loss factor in the model is just η = 0.1 in order to avoid presenting a misleading frequency response. However, in the cases of the stiffened panel, the displacement/velocity amplitudes are more uniformly distributed between that at the edges and the mean amplitudes of the panel. As a result, the validity of the modeling assumptions is reinforced.

The inclusion of the inertial amplifier on the KD mounting system substantially improves the acoustic performance of the system. Figure 4(b) displays that the presence of the added mass further increases the damping capabilities of the panel and establishes a wider and deeper bandgap in the first resonance region. For the sake of comparison, the effect of the panel rigidity is again described for the four aforementioned cases. Similar to the previous comparison, the optimized KD parameters were utilized for each case, but here the IAM mass was included in the optimization process. It is observed that for the non-enhanced panel, the effect of the IAM is barely noticeable, yet even for a small increase in panel stiffness, this effect is evident. The STL curve is shifted toward lower frequencies, while the anti-resonance compensates the detrimental effect of the natural resonance of the simply supported panel.

However, as expected, a slight drawback of this extreme improvement in the resonance region is the following narrow band of reduced attenuation, which is mitigated slightly due to the resonance effect of the internal mass m_D. Another important observation is that the attenuation band increases with increasing rigidity, while this increase is not proportional to the size of the IAM mass. Namely, the optimized mass for the case of f₁ = 10.8 Hz is m_B = 7.6 kg and, at the same time, the width of the attenuated band is almost invisible, while for f₁ = 100 Hz, the IAM mass is m_B = 3.5 kg where a significantly wide attenuated frequency band is observed. Comparing the STL curves of the panels with eigenfrequencies f₁ = 50 and 100 Hz, we see that the latter has an outstanding performance below 140 Hz; nonetheless, above 140 Hz, the former prevails and has a more flattened/uniform response without a large deviation from the FFR panel.

Based on the investigation on the effect of the panel rigidity on the STL, an indicative scenario is studied to show the advantages of the proposed system.

One simple way to increase the panel’s rigidity is by utilizing a grid of box-type section stiffeners, as shown in Fig. 5(a). The width corresponds to the horizontal distance between the upright supporting beams providing the stable frame of the plasterboard.

The selected setup is presented only as an example of a stiffened panel, while other configurations with stiffeners in a cross or “X” layout or utilizing tensegrity structures/mechanisms could prove effective alternatives. In real-life masonry applications, it is a common practice to support the drywall in stable frames, which can further increase the rigidity of the panel. However, this investigation falls out of the scope of the current contribution, the purpose of which is to demonstrate the capabilities of the KD-IAM concept in low-frequency acoustic problems. This is the reason that the added mass of the stiffeners is not taken into account in the STL performance; rather, the stiffeners are seen as just a means of increasing the fundamental eigenfrequency of the panel due to their geometric configuration.

The fundamental eigenfrequency of the panel is derived via modal analysis employing ABAQUS^® assuming simply supported boundary conditions. The panel is discretized by 20-node quadratic solid hexahedral elements, and the stiffeners are discretized by 8-node quadratic doubly curved thick shell elements. The fundamental mode of the panel is depicted in Fig. 5(b) and occurs at f₀ = 90.1 Hz, while the properties of the stiffeners are summarized in Table IV.

It is noted that the FE model has been used only as a means to define the natural frequency of the panel, while the generalized mass and, consequently, stiffness in the analysis are accounted via the assumptions of the LPM. The optimal properties of the KD-IAM are defined by utilizing the optimization algorithm and by assuming hysteretic damping. A conservative loss factor of η = 0.1 has been used; nonetheless, nowadays, someone can easily find industrially produced rubber materials with loss factors up to 0.4.³⁶ The optimized parameters of the KD-IAM mounts are summarized in Table V.

Figure 6 presents the enhanced acoustic performance of the KD-IAM mounted panel, with the black dotted line corresponding to the lumped parameter model as derived from the first mode approximation of the simply supported panel. Comparing the STL curves of the KD-IAM mounted panel with those of the simply supported panel, the mounted panel shows a 110 Hz wide absorption band between 25 and 135 Hz, which corresponds to the resonance region of the simply supported case. Additionally, a maximum STL gain of 32 dB occurs, which, compared to the FFR panel, is a 13 dB increase in STL.

As given in Eq. (43), the mass m_a of the IAM has an amplification factor (cot² θ + 1), which is plotted in Fig. 7 as a function of angle θ. Therefore, for realization of the IAM considering four inertial amplification mechanisms positioned under the panel, a combination of (m_a, θ) needs to be selected in order to provide the desired m_B resulting from the optimization procedure of the KD-IAM model divided in four.

For this particular design, assuming structural steel (⁠$ρ=7800$ kg m⁻³) for the material of the masses m_a with general dimensions 10 × 30 × 35 mm³, the relevant parameters for the IAM mechanism are summarized in Table VI.

Considering four KDamper mounts at each corner of the panel that act in parallel, the optimized parameters given in Table V are divided in four.

In this implementation, the negative stiffness element of the oscillator is realized utilizing disk (Belleville) springs. The exerted force and the equivalent stiffness of this type of springs are non-linear functions of the vertical displacement s of the inner diameter D_i. The height of the spring is denoted by l₀, D_e is the external diameter, and h₀ ≈ l₀ − t, where t is the thickness. The ratio of the outer to the inner diameter of the disk is defined by δ = D_e/D_i. More specific information about disk springs is provided by manufacturers.³⁷ 

Standard disk springs dimensioned in accordance with DIN 2093 have ratios of h₀/t up to 1.3. However, in order for such a disk to demonstrate negative stiffness behavior, this ratio needs to be $h0/t>2$⁠, which is categorized as non-standard.

The range of the vertical displacement s where the disk exhibits the desired negative stiffness characteristics is centered around the flat position of the disk where the exerted force is F_c and the stiffness attains its maximum negative value. This, of course, translates to the application of a certain pre-stress condition on the spring in order to be compressed to its flat position at the equilibrium state of the mount. The height h₀ of the disk spring has to be such that the displacement amplitude C₃ of the internal mass m_D is covered by the resulting range of s corresponding to the negative stiffness values near the maximum within an appropriate margin.

Relevant geometrical parameters are gathered in Table VII. The material properties of the specific disk spring made of ABS plastic are presented in Table VIII.

In order to achieve the required negative stiffness value, $kN/4=−2.193×104$ (N m⁻¹), with the geometrical properties presented in Table VII, a total number of six such disk springs have to be configured in parallel. The resulting stiffness in this case is demonstrated in Fig. 8(a) in comparison with the single spring curve. The bold dashed line indicates the maximum negative stiffness value of k_N, which corresponds to the k_N/4 value from the optimized model.

Figure 8(b) shows the stresses as a function of deformation s at five characteristic points of the spring. Stresses σ_I, σ_II, σ_III, and σ_IV correspond to the points at the four corners of the cross section of the disk, while σ_OM is the equivalent Von Mises stress at the center of the cross section. The value of the resulting stress σ_OM must be between certain limits, which, especially in dynamic applications, defines a nominal number of cycles of operation for the disk spring provided via endurance and fatigue strength diagrams.

The realization of the negative stiffness element via six parallel disks is reconfigured into two stacks of three positioned on a fixed base as demonstrated in Fig. 9 and held together by a bolt and two washers at their flat position. This way, the two stacks still act as two parallel springs, resulting in the desired k_N value; however, the exerted forces of each stack have the same magnitude but opposite directions, meaning that they cancel each other out and the equilibrium position is neutrally stable. The outer and inner diameters of the springs are free to slide on the fixed base and the washers, respectively, while the disks between them can also slide.

The selected material for realization of the positive stiffness elements of the mount k_S/4 and k_P/4 is CR07 (chloroprene rubber with 7% carbon black) in the form of rubber pads. Common rubber materials, such as chloroprene rubbers with carbon black additives, demonstrate the desired damping effects due to internal friction. Typical values range from η = 0.05 up to η = 0.7 depending on the composition and quality of the material. The stiffness of the rubber pads depends on their dimensions.

An indicator of the compressive stiffness of a rubber pad is the shape factor S, which is defined as the ratio of the one loaded surface area A to the area that is free to bulge. Specifically, for hollow cylindrical rubber pads, the shape factor is calculated as

where D and d are the outer and inner diameters, respectively, and t is the thickness of the pad. The compressive stiffness of the rubber pad is calculated as

where

is the corrected Young’s modulus of the material obtained using the calculated shape factor. Obviously, this requires the knowledge of the initial E₀, which can be provided by relevant stress–strain tests of the material in question. For the purpose of this indicative design, the hyperelastic properties of rubber are neglected; consequently, the calculated value from Eq. (55) is an approximation that can be used as a starting point of the detailed dimensioning of the rubber pads. The dimensions of the rubber pads are summarized in Table IX.

Figure 10 shows the model of a KD-IAM unit cell for the simplest case of a meta-structure based on the periodic repetition of such unit cells. The periodicity is considered one dimensional, and the IAMs are grounded on one end to present continuity with the previous part of the analysis as in the KD-IAM mount. However, a 2D perfectly symmetrical lattice would have the same propagation characteristics in both dimensions.

The displacement of a DoF at a certain position 2p of the lattice may be expressed in a complex notation as

with κ being the wavenumber and L being the length of the unit cell.

In the time part of the solution, the parameter σ is defined as σ = ±jΩ, or in the case where the attenuation between the unit cells is considered, $σ=−ζ(κ)Ω(κ)±jωn1−ζ2(κ)$⁠, where ζ is the damping ratio.

Utilizing Bloch’s theory, the propagation of elastic waves between unit cells can be described by considering the interaction of displacements and forces. Considering the spatial part of the solution, this is expressed as

Assuming the case without damping, substitution into the equations of motion of the unit cell leads to

where γ = 2(1 − cos q) for q = 2κL and k_D = k_P + k_N. Setting λ = Ω², Eq. (63) can be written as

where

having defined the following characteristic frequencies and parameters:

Then, the dispersion relations are given by the two solutions λ₁, λ₂ of Eq. (64), resulting in the upper $ω+(q)=λ1$ and lower $ω−(q)=λ2$ branches, respectively.

The dispersion relations are 2π/L-periodic in the wavenumber space; therefore, ω(q) = ω(q + 2π). Furthermore, considering the irreducible Brillouin zone, the two characteristic high (ω_H) and low (ω_L) frequencies of the generated bandgap can be calculated as

where $ρB=ωDωB$⁠. Additionally, the “central” frequency of the bandgap, in the case where without damping wave propagation is completely prohibited, is calculated as

where κ_D = k_D/k₀. The above frequencies can be used for tuning of the meta-structure for a specific application and for the definition of the normalized bandgap width in the form b_w = (ω_H − ω_L)/ω_L.

Using the same parameter values as calculated for the previous case of the KD-IAM mounts, the bandgap frequencies, shown in Fig. 11(a), of the irreducible Brillouin zone are obtained as f_H = 163.32 Hz and f_L = 56.67 Hz, corresponding to a normalized bandgap width b_w = 1.89.

Figure 11(b) demonstrates the acoustic performance of a corresponding acoustic panel based on the periodic repetition of KD-IAM unit cells in comparison to the indicative implementation of Sec. IV. For the purpose of this comparison, the previously defined modal mass m₁ is divided in two masses enclosing the periodic chain in both ends. This modeling technique is required to simulate the effect of the radiating surface and the consequent calculation of the STL. The acoustic performance is then evaluated for various numbers of unit cells.

The comparison shows that on increasing the number of unit cells while maintaining the mass of the radiating surfaces, the meta-structure provides improved damping, especially in the region of the lower resonances. It should be noted that for this simulation of the meta-structure, the loss factor of the stiffness elements (k_S′, k_P′) has been chosen as η = 0.02, which is five times lower than in the case of KD-IAM mounts. The depth and width of the attenuation band are not significantly affected as far as the STL performance is concerned. However, despite the choice of lower loss factor, the emergence of meta-damping improves the response around the characteristic frequency f₀, which can be very important for certain applications. However, in the context of the present investigation, it is demonstrated that the physical mechanisms of the KD-IAM concept providing this extreme attenuation band are present even for a single unit cell, meaning that, at least in this specific case, the added manufacturing complexity for multiple unit cells could be deemed unnecessary.

Figure 12 shows a feasible conceptual design for such a meta-structure for M = 1 number of unit cells according to the convention used in this section. Naturally, the various parallel KD-IAM elements can be divided according to the number of positions and such unit cells are chosen to be positioned on the surfaces of the panels. In addition, such a structure would be required to be in a pre-stressed state. Namely, in this indicative design where the negative stiffness elements are realized with curved bistable beams, the pre-stress is needed to drive the beams in their post-buckling state. This also means that the total thickness of the meta-structure will be reduced due to the consequent compression of the internal elements.

The performance of widely used structural elements for passive solutions of acoustic treatment such as plasterboard panels, regarding the mitigation of low-frequency noise, relies largely on the attenuation of the fundamental resonance region dynamic response.

Since the boundary conditions of a vibrating panel affect its dynamic response, the sound transmission loss (STL) in this frequency range can be manipulated by supporting the panel on elastic mounts. The presented implementation utilizing the proposed KD-IAM advanced absorber for the mounting of the panel shows a resulting wide and deep frequency band of improved vibration and noise attenuation in terms of STL in the resonance region and at frequencies of ∼20–150 Hz.

The optimization procedure for selection of the absorber’s parameters based on the described modeling approximations demonstrated the relation between the rigidity and the STL performance of the mounted panel. Utilizing stiffeners to increase the rigidity and, consequently, the fundamental frequency of the panel, the resulting optimized mounts based on the KD-IAM achieve a wider and deeper frequency band of improved STL in the resonance region of the system.

Next, an indicative realization of the KD-IAM based mounting system is presented utilizing Belleville springs for the negative stiffness element of the KDamper and other common elements, such as rubber pads and simple linkages.

Finally, a meta-structure based on the periodic repetition of KD-IAM unit cells demonstrates an improvement in the damping capacity, especially at low-frequency resonances for increasing number of unit cells. However, the results also indicate that the general improvements in STL performance are not proportional to the number of unit cells. Thus, the KD-IAM is deemed extremely capable for low frequency acoustic insulation without the requirement of many unit cells and the added complexity that could be entailed in a practical implementation.

M.K. has been financed by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant (Grant Agreement No. INSPIRE-813424, “INSPIRE—Innovative Ground Interface Concepts for Structure Protection”).

The data that support the findings of this study are available within the article and from the corresponding author upon reasonable request.