- Email: [email protected]
Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures Efe Ozkaya, Cetin Yilmaz n Department of Mechanical Engineering, Bogazici University, 34342, Bebek, Istanbul, Turkey
a r t i c l e i n f o
abstract
Article history: Received 11 April 2016 Received in revised form 31 October 2016 Accepted 16 November 2016 Handling Editor: A.V. Metrikine
The effect of eddy current damping on a novel locally resonant periodic structure is investigated. The frequency response characteristics are obtained by using a lumped parameter and a finite element model. In order to obtain wide band gaps at low frequencies, the periodic structure is optimized according to certain constraints, such as mass distribution in the unit cell, lower limit of the band gap, stiffness between the components in the unit cell, the size of magnets used for eddy current damping, and the number of unit cells in the periodic structure. Then, the locally resonant periodic structure with eddy current damping is manufactured and its experimental frequency response is obtained. The frequency response results obtained analytically, numerically and experimentally match quite well. The inclusion of eddy current damping to the periodic structure decreases amplitudes of resonance peaks without disturbing stop band width. & 2016 Elsevier Ltd All rights reserved.
Keywords: Phononic band gap Periodic structure Eddy current damping Local resonator
1. Introduction The frequency ranges, in which propagation of acoustic or elastic waves are hindered, are called phononic band gaps [1– 5]. Various methods can be used to generate phononic band gaps. In Bragg scattering method, the interference of the incident and reflected waves attenuates the energy of the propagating wave to create phononic band gaps [6,7]. In inertial amplification method, the wave propagating medium's effective inertia is amplified via embedded amplification mechanisms [3,6,7]. In local resonance method, when the frequency of the incident wave upon the periodic structure is close to the resonance frequency of the resonators, the wave energy excites the resonators and wave propagation is hindered. As a result, a band gap is formed around the resonance frequencies of the local resonators [8–13]. When there is no damping in a locally resonant structure, deep anti-resonance notches can be observed in its frequency response function enabling excellent vibration isolation around these frequencies [14]. However, in an undamped structure with local resonators, sharp resonance peaks are observed. Consequently, if the structure is excited around these frequencies, large undesired amplifications can be seen. Former studies have considered the effect of damping in lattices with local resonators [15–17]. Generally, viscous damping is investigated in these studies. In this paper, eddy current damping will be included to the local resonators in a periodic structure with the aim of decreasing the amplitudes of the resonance peaks while preserving a deep and wide stop band. Eddy currents are created when a non-magnetic conductor is moving in a region imposed upon a stationary magnetic field or when a time varying magnetic field has an impact upon a stationary non-magnetic conductor. According to Lenz's n
Corresponding author. E-mail addresses: [email protected] (E. Ozkaya), [email protected] (C. Yilmaz).
http://dx.doi.org/10.1016/j.jsv.2016.11.027 0022-460X/& 2016 Elsevier Ltd All rights reserved.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Law, eddy currents circulating in a conductor create their own magnetic field. As a result, a resistive force which opposes the motion of the moving part is created. When vibration between a conductor and a magnet is considered, a damping force proportional to the velocity of the vibration is generated and this is called eddy current damping [18–25]. Both viscous and eddy current damping forces are proportional to the velocity of the vibration. Therefore, mathematically these two damping mechanisms are equivalent. However, when the physical structure of these dampers are considered, one can see that eddy current damping offers certain advantages. In viscous dampers, generally a fluid is forced through a small opening by a moving piston. Due to sliding motion, there may be wear problems. Moreover, compressibility of the fluid inside these dampers introduce some stiffness besides damping [26]. Stiffness change due to compressibility effect is more pronounced at high frequencies. If eddy current damping is added to a structure, its stiffness is not disturbed as eddy current damping is a non-contact type of damping. Due to its non-contact nature, it is not prone to wear. Eddy current damping method has been used in various applications. Sodano et al. [27] used eddy current damping to impede the transverse vibrations of a cantilever beam. Ebrahimi et al. [28] studied a design that involves two permanent magnets and a conductive aluminum plate to generate both stiffness and variable damping. Elbuken et al. [29] applied eddy current damping to magnetically levitated miniaturized objects. Bae et al. [30] introduced a tuned mass damper with eddy current damping. In this paper, eddy current damping will be combined with the local resonance method for the first time. In order to generate eddy current damping, concentric copper tube and magnet assemblies will be used. To enforce one-dimensional (1D) motion between these components, a novel periodic structure with parallel spiral springs will be designed. Parametric studies will be conducted with the aim of obtaining a wide local resonance induced gap. The periodic structure will be manufactured to see the effect of eddy current damping on resonance peaks, stop band width, and depth in frequency response plots.
2. Models and methodology The study consists of three main sections. Firstly, analytical calculations are done to determine the values of the design parameters. Depending on these parameter values, a prototype design is established numerically and modal analysis is conducted. In the final phase, experimental validation is performed for the prototype produced. In this study, small deformations are considered, hence the structure is assumed to be linear and strength is not an issue. 2.1. Analytical model of the locally resonant periodic structure A one-dimensional locally resonant periodic structure that includes eddy current damping is modelled as in Fig. 1. During modelling of the periodic structure, five parameters are taken into account, which are resonator mass (mr), total mass of the remaining parts in the unit cell when the resonator mass is excluded (m), stiffness of the spring connecting the resonator mass to the unit cell (kr), total stiffness between the unit cells (k) and the eddy current damping constant (c). As eddy current damping force is proportional to the velocity of vibration, c can be considered as a viscous damping constant. Phononic band structure and the frequency response of the periodic structure will be determined by the transfer matrix method [31]. Through mechanical impedance to mass conversion, periodic structure's effective mass can be calculated to be used in its overall transfer matrix. Mechanical impedance representations for mass, spring and damper elements are jωm, k/jω and c, respectively [32]. By using these impedances, effective mass of the unit cell of the system shown in Fig. 1(b) is obtained as follows:
meff = m +
1 1 ω2 − mr k r + jωc
(1)
The point transfer matrix for meff (Eq. (2)) and the field transfer matrix for k (Eq. (3)) are used [31] to obtain the relationship between consecutive unit cells (Eq. (5)) through the state vector relationship as shown in Eq. (4).
Fig. 1. Lumped parameter model of the locally resonant periodic structure including eddy current damping (a) 1D array with local resonators, (b) the equivalent array with effective mass.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
⎡ 1 0⎤ ⎥ Pi = ⎢ 2 ⎣ − meff ω 1⎦
(2)
⎡ 1⎤ ⎢1 ⎥ Fi = ⎢ k⎥ ⎣ 0 1⎦
(3)
⎡ u ⎤R ziR = ⎢ ⎥ ⎣ N ⎦i
(4)
ziR = Pi Fi ziR− 1
(5)
where u is displacement and N is internal force. To construct the band structure of the locally resonant periodic structure that includes eddy current damping, Bloch's Theorem (Eq. (6)) is applied [33].
ziR = ziR− 1e jγ
(6)
Here, γ represents the wave number. Eqs. (4), (5) and (6) can be used to obtain the relationship for a unit cell with periodic boundary conditions (Eq. (7)).
⎡ 1⎤ R 1 0⎤ ⎢ 1 ⎡ u ⎤R ⎡ ⎥⎡ u ⎤ k ⎥ ⎢⎣ ⎥⎦ e−jγ ⎢⎣ N ⎥⎦ = ⎢ − m ω2 1⎥ ⎢ N ⎣ ⎦ eff i i ⎣0 1⎦
(7)
Due to periodic boundary conditions, the determinant expressed in Eq. (8) is obtained.
1 − e jγ
1 k
− meff ω2
−meff ω2 + 1 − e jγ k
=0 (8)
Consequently, dispersion equation (Eq. (9)) for the periodic structure including infinite number of unit cells is attained. ω values that satisfy this dispersion equation are obtained for γ values in between 0 and π. As a result, the five aforementioned design parameters can be optimized for a desired band gap width.
1 − 2e jγ + e2jγ +
meff ω2e jγ =0 k
(9)
On the other hand, to obtain the transmissibility plot for the periodic structure with n unit cells, overall transfer matrix of the structure should be calculated by using Eqs. (2) and (3). In the overall transfer matrix, the U22 entry is used to find the displacement or acceleration transmissibility (TR (ω))(see Eq. (11))
⎡ U11 U12 ⎤ Uoverall = (Pi Fi )n = ⎢ ⎥ ⎣ U21 U22 ⎦ TR (ω) =
n = 1, 2, 3, …
x (ω) x¨ (ω) 1 = = y (ω) y¨ (ω) U22
(10)
(11)
where x (ω) and y (ω) are the output and input displacements. Moreover, the frequency range where TR (ω) < 1 is defined as the stop band. In order to show the effect of eddy current damping on the frequency response of the periodic structure, its lumped parameter model is optimized to obtain a stop band between 50 and 100 Hz frequency range. During optimization, parametric studies are conducted by using the dispersion equation (Eq. (9)) and the transmissibility (Eq. (11)) of the lumped parameter model shown in Fig. 1. 2.2. Finite element models of the periodic structure Eddy current damping within the periodic structure will be realized through ring magnet and copper tube assemblies. The magnets will be constrained to move concentrically inside the copper tubes with small amplitudes. Hence, orientation of the magnetic field lines generated by each magnet or magnetic flux density within the corresponding copper tube will change negligibly during vibration. Consequently, the damping coefficient generated by each magnet can be assumed as constant for small amplitude vibrations. Electromagnetic finite element models of these assemblies will be formed to calculate their damping coefficients. Moreover, structural finite element models of the periodic structure will be generated Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Fig. 2. Band structure plots for three different mass ratios (a) m/mr = 1/2, (b) m/mr = 1/3, (c) m/mr = 1/4 . In all plots m = 0.1 kg , k ¼ 126 kN/m, c ¼0 N s/m and kr /mr = 2π × 50 rad/s.
to be used for modal analysis. The equivalent viscous damping coefficients of the magnet and copper tube assemblies will be used in the structural finite element and lumped parameter models. Finally, frequency response results of the lumped parameter and finite element models will be compared.
3. Parametric studies First two parametric studies focuses on the effect of mass distribution within the unit cell and the number of unit cells to the width and depth of the stop band of the periodic structure. While the latter two focuses on the effect of eddy current damping to the frequency response of the periodic structure. First of all, the effect of remaining mass of the unit cell when the resonator is excluded (m) to resonator mass (mr) ratio is investigated. In this study, two different ring magnets with masses 0.032 kg and 0.052 kg are planned to be used for eddy current damping purposes. Thus, the remaining mass (m) of the unit cell including these magnets is predicted to be around 0.1 kg . On the other hand, the resonator mass (mr) which mainly consists of copper tube can be varied to change the band gap frequency range. Fig. 2 shows that as m /mr ratio decreases, wider band gaps can be obtained. It is seen that m /mr = 1/3 provides a band gap in the target frequency range (50–100 Hz). Secondly, the effect of number of unit cells to the width and depth of the stop band is investigated. Fig. 3 shows that as the number of unit cells is increased, anti-resonance depth at 50 Hz also increases. Therefore, vibration isolation performance in between 50 and 100 Hz is improved. However, as the number of unit cells increases, the overall size and mass of
Fig. 3. The effect of number of unit cells in the undamped periodic structure on the depth of the stop band between 50 and 100 Hz. Here, m = 0.1 kg , mr = 0.3 kg , k ¼126 kN/m, kr ¼ 29.6 kN/m.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Fig. 4. The effect of damping ratio of the resonators on stop band width, resonance peaks and anti-resonance depth for a periodic structure with four unit cells. Here, m = 0.1 kg , mr = 0.3 kg , k ¼126 kN/m, kr ¼ 29.6 kN/m.
the periodic structure increases, as well. Considering the compromise between stop band depth and mass of the periodic structure, four unit cells are chosen to be used in the design. Within the unit cell of the periodic structure, a ring magnet will be used inside a copper tube to generate eddy current damping. In the third parametric study, the effect of damping ratio on the stop band width and depth of the structure will be investigated. Only the eddy current damping constant or equivalent viscous damping coefficient (c) will be varied in the lumped parameter model of the periodic structure. Fig. 4 shows that as the damping ratio of the resonators (see Eq. (12)) increases, all the resonance peaks and the anti-resonance depth at 50 Hz decreases.
ζ=
c 2 mr k r
(12)
Also note that the increase in the damping ratio slightly increases the stop band width. However, if larger damping coefficients are to be realized by the use of larger magnets in the design, then the mass of the unit cell excluding the resonator (m) will increase. Consequently, m /mr ratio will increase, resulting in a narrower stop band as shown in Fig. 2. In the literature, many parametric studies are conducted regarding cylindrical magnets moving inside copper tubes. It has been shown that eddy current damping force or Lorentz force is affected by the cylindrical magnet's radius, height, and grade; copper tube's height, wall thickness and inner diameter (or the air gap between the tube and the magnet) [19,20,24,25,34,35]. Similarly, eddy current damping force or Lorentz force generated by a ring magnet can be affected by the magnet dimensions and grade, and copper tube dimensions. Two set of NdFeB ring magnets (see Table 1) will be used in the locally resonant periodic structure to generate eddy current damping. Notice that the remanent flux densities of these N35 grade magnets can be in a quite wide range depending on the manufacturer. In order to determine the remanent flux densities of these magnets accurately, free fall experiments are conducted. To that end, a 550 mm long 1.5 mm thick copper tube with 32 mm inner diameter is used in which the ring magnets are let to free fall. The free fall times of the ring magnets are measured with a digital chronometer. The average of 10 measurements are recorded. In line with the literature [19,20,24,25,34,35], the acceleration times of the magnets are neglected and the magnets are assumed to descent with terminal velocity during their entire fall. Later, the length of the copper tube is divided by these free fall times and terminal velocities are obtained. Since at terminal velocity (vterminal) Lorentz force (Florentz) is equal to the weight of the magnet (mg), by dividing the Lorentz force by the terminal velocity, damping constant (c) can be obtained as in Eq. (13).
c=
Florentz mg = vterminal vterminal
(13)
Following this experiment, an electromagnetic finite element model which is a duplicate of this experiment is prepared. During the finite element analysis, remanent flux densities of each ring magnet are varied from 0.9 T to 1.5 T with 0.01 T increments to get the same terminal velocity values as in the free fall experiment. In Table 2, corresponding remanent flux Table 1 Dimensions, weights and remanent flux density intervals of the ring magnets.
Magnet outer radius (mm ) Magnet inner radius (mm ) Magnet height (mm ) Magnet weight (N ) Remanent flux density interval (T)
Small magnet
Large magnet
12.5 4.25 10 0.31 0.9–1.5
15.35 4.25 10 0.51 0.9–1.5
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Table 2 Free fall experiment results for ring magnets. Here, the copper tube is 550 mm long, 1.5 mm thick and its inner diameter is 32 mm.
Measured free fall time (s) Terminal velocity (m/s) Lorentz force (N) Damping constant (N s/m) Remanent flux density (T)
Small magnet
Large magnet
3.18 0.173 0.31 1.81 1.03
9.59 0.057 0.51 8.90 1.14
Fig. 5. Computational analysis for the free fall of the large ring magnet inside a copper tube with 11 mm wall thickness and 23.2 mm height. (a) Section view, (b) 2D axi-symmetric meshed geometry, (c) magnetic flux density norm (T) contour plot, (d) current density norm (A/m2 ) contour plot, (e) relative velocity between magnet and copper tube, and (f) Lorentz force between magnet and copper tube that is generated due to eddy currents within the copper tube.
densities are given. A final parametric study will be conducted to obtain the maximum amount of damping for the given ring magnets. To that end, an electromagnetic finite element model of the ring magnets and copper tube will be generated. The magnets will Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
Fig. 6. The effect of copper tube's wall thickness on damping behaviour for a constant resonator mass of 0.3 kg . (a) Effect of copper the wall thickness on damping constant, (b) effect of copper tube wall thickness on damping ratio. Here, mr = 0.3 kg , kr ¼29.6 kN/m.
be subjected to free fall due to gravity and the Lorentz force between magnets and copper tubes will be calculated. During the analyses, mass of the copper tube is kept constant at 0.3 kg . Moreover, its inner diameter is fixed at 32 mm while its height and wall thickness is varied considering the mass constraint. Height of the tube is decreased as the wall thickness is increased from 1 mm to 20 mm with 1 mm increments. One of these analyses is shown in Fig. 5. Here, the wall thickness of the copper ( ρ = 8700 kg/m3) tube is 11 mm, and its height is 23.2 mm satisfying the mass constraint of 0.3 kg . Notice that as the magnet velocity reaches a terminal value, the Lorentz force in the direction of free fall (z-direction) reaches 0.51 N, which is the weight of the large magnet. Then, damping constant can be found using Eq. (13). For this case, vterminal = 0.0167 m/s giving c¼30.5 N.s/m. Fig. 6 shows the effect of wall thickness of the copper tube on the eddy current damping constant and the damping ratio of the resonator. It is seen that to acquire the maximum damping ratio under a mass constraint, there is an optimum copper wall thickness value. For a copper tube having an inner diameter of 32 mm with a mass constraint of 0.3 kg , the optimum wall thickness value is around 11 mm.
4. Design and finite element analysis of the locally resonant periodic structure In Fig. 7, lumped parameter model of the unit cell of the periodic structure and the corresponding distributed parameter model can be seen. In Fig. 7(b) two sets of parallel spiral springs are used to realize the stiffnesses k and kr. The two parallel spiral springs (k r /2) that are attached to the copper tube (mr) allow the ring magnet to move concentrically inside the tube without any contact. Notice that as the large magnet diameter is 30.7 mm and the inner diameter of the tube is 32 mm, there is very little clearance between them. The other two parallel spiral springs (k/2) are used to connect the unit cells with each other. Due to their parallel design, they mainly allow axial relative motion between the unit cells. As a result, a novel onedimensional locally resonant periodic structure can be realized with prescribed stiffness characteristics for axial excitations. In the analyses with the lumped parameter model, the stiffnesses kr and k are determined as 29.6 kN/m and 126 kN/m (see Figs. 3 and 4). In order to realize these stiffness values, the spiral springs shown in Fig. 8(a) and 8(b) are designed. Both of these stainless steel (E ¼193 GPa, ρ = 7750 kg/m3, ν = 0.31) springs have 60 mm outer diameter, 8.5 mm inner diameter and 0.7 mm thickness. The spiral shaped slots are chosen such that the stiffness of these springs are close to half of the stiffness values in the lumped parameter model. The locally resonant periodic structure with four unit cells can be seen in Fig. 9. The whole structure has an outer diameter of 60 mm and length of 376 mm. Notice that the central stainless steel M8 bolts connect all parts within each unit cell. The bolt head is connected to a stainless steel circular plate that has 1 mm thickness. Moreover, three stainless steel M4
Fig. 7. (a) Lumped parameter model of the unit cell. (b) Distributed parameter model of the unit cell.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
Fig. 8. (a) Spiral spring with kr /2 = 13.5 kN/m stiffness. (b) Spiral spring with k/2 = 62.0 kN/m stiffness.
Fig. 9. The locally resonant periodic structure with four unit cells. (a) Isometric view, (b) side view.
bolts are used near the periphery of each circular plate to make the connection with the neighbouring unit cell. Small plastic rings and nuts with negligible masses will be used during the assembly of the locally resonant periodic structure. However, as the effect of these plastic parts are assumed to be negligible, they are not included in the model shown in Fig. 9. In the parametric studies with the lumped parameter model, m and mr were taken as round numbers like 100 g and 300 g. When the distributed parameter model of the structure is formed, m is calculated as 0.143 kg for the unit cell with small magnet. However, mr is 0.392 kg giving m /mr = 0.365 which is close to the target mass ratio of 1/3. Furthermore, m becomes 0.162 kg and 0.111 kg for the unit cell with large magnet and no magnet. As a result, mass ratio for the large magnet and no magnet cases are 0.413 and 0.283, respectively. The copper tube with mass mr in Fig. 7(b) does not have constant wall thickness as in Fig. 5(a). Notice that the wall thickness is smaller on the two faces allowing room for out-of-plane deformations of the spiral springs (k r /2). The wall thickness in the middle cross-section is chosen as 14 mm, which is a little higher than the optimum value of 11 mm to compensate the decrease in the wall thickness on the two faces. In Fig. 10, electromagnetic finite element analysis of the designed copper tube shape can be seen. Here, the large magnet is shown inside the tube. The same analysis is conducted for the small magnet, as well. As a result of these analyses, the damping constants for the small and large magnets are found as 4.02 N.s/m and 20.4 N.s/m, respectively. These values are a little lower than the peak values in Fig. 6(a). When the eddy current density plots in Fig. 5(d) and Fig. 10(d) are compared, higher current densities are seen over larger areas in Fig. 5(d). This explains the lower damping constants for the designed copper tube. The relationship between magnetic flux density (B) and magnetic field (H) is given as
B = μH
(14)
where μ represents the permeability of the medium [36]. Therefore, a change in magnetic field results in a change in magnetic flux density as μ is constant. If magnetic materials like the stainless steel parts in the resonator are present near a magnet, then they can distort the magnetic field lines. As a result, magnetic flux density can change. In order to check the effect of other elements within the resonator on the damping behaviour, a more detailed model of the resonator is analysed. In Fig. 11, the stainless steel (σ = 1.32 × 106 Siemens/m ) spiral springs k r /2, stainless steel M8 bolt in the middle and the plastic rings that are used to separate the magnet and the spiral springs are included in the model. As a result of this analysis, the damping constants for the small and large magnets are found to be 2.31 N.s/m and 10.8 N. Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
9
Fig. 10. (a) Section view of a large ring magnet inside the designed copper tube. (b) 2D axi-symmetric finite element model of the designed copper tube and large ring magnet. (c) Magnetic flux density norm (T) contour plot. (d) Current density norm (A/m2) contour plot.
Fig. 11. (a) Section view of the fully represented resonator. (b) Fully represented 2D axi-symmetric resonator geometry with large magnet included. (c) Magnetic flux density norm (T) contour plot. (d) Current density norm (A/m2) contour plot.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
10
Fig. 12. Finite element mesh for the locally resonant periodic structure with small magnets included. The resonator of the second unit cell is suppressed for visual purposes.
s/m, respectively. The smaller damping constant values in this analysis can be explained by observing that the stainless steel bolt and stainless steel spiral springs reduce magnetic flux density close to the copper tube (compare Figs. 10(c) and 11(c)). Consequently, smaller eddy current densities are obtained in Fig. 11(d) when compared to Fig. 10(d). Notice that the interaction between magnets is not considered in this study since the magnets are far away from each other as shown in Fig. 9. However, if the unit cell length was smaller, then each magnet would interact with the neighbouring magnets which in turn would change the stiffness characteristics of the unit cell. In that case, the magnetic field lines of each magnet would also interact with the neighbouring copper tubes that would change the damping characteristics, as well.
4.1. Structural finite element analyses of the locally resonant periodic structure The finite element model of the locally resonant periodic structure shown in Fig. 9 is formed and its first 10 natural frequencies are determined for three cases: resonators with no magnets, resonators with small magnets, and resonators with large magnets. The finite element mesh for the case with small magnets can be seen in Fig. 12. Similar mesh sizes are used in the other two cases. In all cases, the structures are fixed from the right side and frictionless supports are used around the resonators. Table 3 shows the first 10 natural frequencies of the locally resonant periodic structure without any magnets. When the corresponding mode shapes are investigated, it is seen that the 5th mode is torsional as shown in Fig. 13(b). Therefore, this mode is not expected to be seen in the frequency response if the system is excited axially. As a result, the stop band will occur between the 4th and 6th modes, which are shown in Figs. 13(a) and 13(c). Second modal analysis is done when small ring magnets are inserted inside the copper tubes. The first 10 natural frequencies are shown in Table 4. The stop band will occur in between 4th and 6th modes and they are depicted in Fig. 14 with the torsional mode in between at 62.9 Hz. These limiting mode shapes are shown in Figs. 14(a) and 14(c). Finally, a modal analysis is done when large ring magnets are inserted inside the copper tubes. The first 10 natural frequencies are shown in Table 5. The stop band will again occur in between 4th and 6th modes with a torsional mode in between and they are shown in Fig. 15.
5. Experimental measurements and results 5.1. Laser vibrometer measurements to estimate the eddy current damping constants To validate the damping constant values obtained through the electromagnetic finite element analysis, measurements are taken via a laser vibrometer for the three types of resonators. Fig. 16 shows one of these resonators. In the first measurement no ring magnet is inserted inside the copper tube. In the second and third measurements, small and large ring magnets are inserted inside the copper tube to create eddy current damping. The damping coefficient in all three case are calculated through Eq. (15).
c = 2ζ meff k r
(15)
In the experiments, the copper tube is fixed and the inner part is allowed to vibrate. Thus, effective mass (meff) used in Eq. (15) differs depending on the mass of the magnet used inside the resonator (see Table 6). In Eq. (15), kr ¼27 kN/m and ζ is determined by the logarithmic decrement method. Generally, this method can be formulated as follows [37]: Table 3 First 10 natural frequencies of the periodic structure without any magnets. In the last row, types of the modes are indicated as axial (A) or torsional (T). Mode number Frequency (Hz) Mode type
1 30.7 A
2 45.2 A
3 47.2 A
4 47.7 A
5 66.2 T
6 130.8 A
7 170.5 T
8 231.5 T
9 257.6 A
10 258.3 T
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
11
Fig. 13. Mode shapes of periodic structure without any ring magnets that determine the lower and upper limits of the stop band with a torsional mode in between. (a) 4th mode shape at 47.7 Hz , (b) 5th mode shape at 66.2 Hz and (c) 6th mode shape at 130.8 Hz . Table 4 First 10 natural frequencies of the periodic structure including small ring magnets. In the last row, types of the modes are indicated as axial (A) or torsional (T). Mode number Frequency (Hz) Mode type
δ=
ζ=
1 28.7 A
2 43.1 A
3 44.6 A
1 ⎛ x (t ) ⎞ ln ⎜ ⎟ n ⎝ x (t + nT ) ⎠
4 46.5 A
5 62.9 T
6 111.3 A
7 163.2 T
8 215.4 T
9 220.1 A
10 244.3 T
(16)
δ 4π 2
+ δ2
(17)
where δ is the logarithmic decrement, x(t) is the peak displacement amplitude at time t, x (t + nT ) is the peak displacement amplitude after n periods (T). Three different measurement results are shown in Fig. 17 with six data points taken in order to apply the logarithmic decrement method given in Eqs. (16) and (17). Notice that in Eq. (16) displacement amplitude information is used. However, vibration velocity is measured by the laser vibrometer. Therefore, Eq. (16) needs to be modified. Since this equation is derived by assuming that the motion is underdamped, displacement amplitudes are related by Eq. (18)
x (t + nT ) = e−ζω n nT x (t )
(18)
where ωn is the undamped natural frequency of the system. When we take the time derivative of Eq. (18) the following equation is obtained:
v (t + nT ) = e−ζω n nT v (t )
(19)
where v(t) is the velocity amplitude at time t, v (t + nT ) is the velocity amplitude after n periods (T). There is a phase difference between peak velocity and peak displacement values. Nevertheless, for all values of t, the following relationship holds:
x (t )/x (t + nT ) = v (t )/v (t + nT ) = e ζω n nT
(20)
Therefore, Eq. (20) can be used for the peak values in the velocity measurements. As a result, Eq. (16) can be reformulated as Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
12
Fig. 14. Mode shapes of periodic structure with small magnets that determine the lower and upper limits of the stop band with a torsional mode in between. (a) 4th mode shape at 46.5 Hz , (b) 5th mode shape at 62.9 Hz and (c) 6th mode shape at 111.3 Hz . Table 5 First 10 natural frequencies of the periodic structure including large magnets. In the last row, types of the modes are indicated as axial (A) or torsional (T). Mode number Frequency (Hz) Mode type
1 28.9 A
2 43.0 A
3 45.1 A
4 46.7 A
5 64.5 T
6 106.1 A
7 164.3 T
8 202.5 T
9 221.8 A
10 245.5 T
Fig. 15. Mode shapes of periodic structure with large magnets that determine the lower and upper limits of the stop band with a torsional mode in between. (a) 4th mode shape at 46.7 Hz , (b) 5th mode shape at 64.5 Hz and (c) 6th mode shape at 106.1 Hz .
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
13
Fig. 16. Measured resonator with a small ring magnet inside.
Table 6 Calculated damping ratios and constants via logarithmic decrement method.
Time (s) Velocity (m/s) δ ζ meff (kg) kr (kN/m) c (N s/m)
No magnet
Small magnet
Large magnet
7th peak
15th peak
7th peak
14th peak
3rd peak
6th peak
0.0423 0.0396 0.0387 0.0062 0.034 27.0 0.37
0.0958 0.0291
0.0544 0.0142 0.1537 0.0245 0.066 27.0 2.06
0.1158 0.00485
0.0225 0.00716 0.6882 0.1088 0.085 27.0 10.43
0.0529 0.000909
Fig. 17. Eddy current damping effect measured with laser vibrometer.
δ=
1 ⎛ v (t ) ⎞ ln ⎜ ⎟ n ⎝ v (t + nT ) ⎠
(21)
In Table 6, peak points taken for each measurement to be used in logarithmic decrement method are given, which are followed by the calculated damping ratios and damping constants. If the damping constant of the no magnet case is subtracted from the other cases, eddy current damping constants for each ring magnet can be obtained. Notice that these damping constant values are close to the values obtained through the fully represented resonator model (Fig. 11). Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
14
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 18. Experimental setup.
5.2. Experimental modal analysis of the prototype The prototype of the locally resonant periodic structure and the experimental setup can be seen in Fig. 18. Notice that the prototype is hanged by rubber cords to simulate free boundary conditions. Excitation is given from one end by a modal shaker. The input acceleration is measured through an impedance head and the output acceleration on the other end of the prototype is measured by an accelerometer. Transmissibility is calculated as the ratio of output acceleration to input acceleration (see Eq. (11)). In the three measurements taken, modal shaker excites the structure up to 400 Hz. In the first measurement, to observe the effect of structural damping within the prototype on the vibration stop band, ring magnets are not inserted inside the copper tubes. During the second and third measurements, small and large ring magnets are used, respectively. The effect of eddy current damping can be seen in Fig. 19 via the decrease in all resonance peaks. The stop band width does not change much due to magnet usage. For comparison purposes, frequency response plots are obtained analytically and numerically using the lumped parameter and finite element models, respectively. In Fig. 20, frequency response plots for the lumped parameter models can be seen. From the analytical calculations done by using the lumped parameter model without ring magnets, stop band is obtained in between 40.1 and 88.8 Hz for mr = 0.392 kg , m = 0.111 kg , k¼ 124 kN/m and kr ¼27 kN/m. When small ring magnets are included, stop band is obtained in between 38.7 and 80.9 Hz for mr = 0.392 kg , m = 0.143 kg , k ¼124 kN/m and kr ¼27 kN/m. It is seen that only the m value changes due to small ring magnet inclusion. This causes a decrease in stop band width due to mass ratio (m /mr ) increase. Finally, when the large ring magnets are inserted inside the copper tubes, stop band is obtained in between 37.4 and 78.8 Hz for mr = 0.392 kg , m = 0.162 kg , k¼ 124 kN/m and kr ¼27 kN/m. Inclusion of large ring magnets with higher damping decreases the stop band width. On the other hand, the resonance peaks are suppressed the most with the use of large ring magnets. Hence, there is a compromise between stop band width and damping. Fig. 21 shows the frequency response plots for the finite element models. Notice that the frequency response plots obtained analytically, numerically and experimentally are quite similar (see Figs. 19–21). There are small peaks around 60 Hz and 160 Hz in the numerical frequency response plot due to the torsional modes (see Tables 3–5). However, these peaks are insignificant and do not change the stop band characteristics. In Table 7, lower and upper stop band limits, and their ratios are presented. One can see that there are some shifts in the stop band frequency limits obtained through three different approaches. Nevertheless, ωu/ωl ratios are almost the same.
Fig. 19. Frequency response plots obtained from the experimental measurements.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
15
Fig. 20. Frequency response plots obtained via the lumped parameter models.
Fig. 21. Frequency response plots obtained via the finite element models.
Table 7 Lower (ωl) and upper (ωu) stop band limits, and their ratios (ωu/ωl ) obtained from the analytical, numerical and experimental FRF plots. Here, all frequencies are in Hz. No magnet
Analytical Numerical Experimental
Small magnet
Large magnet
ωl
ωu
ωu/ωl
ωl
ωu
ωu/ωl
ωl
ωu
ωu/ωl
40.1 47.7 52
88.8 105.2 110
2.21 2.20 2.11
38.7 43.6 46.1
80.9 92 100.4
2.09 2.11 2.18
37.4 38.2 39.1
78.8 89.3 100.4
2.11 2.34 2.57
6. Conclusions In this paper, a novel one-dimensional locally resonant periodic structure is designed to generate a stop band at low frequencies. In order to analyse the periodic structure, a lumped parameter and a finite element model are generated. Within the unit cell of the periodic structure, two sets of spiral springs are used. Axial relative motion between the unit cells is accomplished by the parallel placement of these spiral springs. Parametric studies regarding the effect of mass distribution within the unit cell and the number of unit cells on the stop band characteristics are conducted. It is seen that when the unit cell mass excluding the resonator to resonator mass ratio is about one-third, the targeted stop band width is achieved. As the number of unit cells increases, deeper stop bands are obtained with the burden of increasing the overall size and mass of the periodic structure. Considering the compromise between stop band depth and mass of the periodic structure, four unit cells are chosen. Furthermore, eddy current damping effect is included to the periodic structure through resonator masses being made of copper tubes in which ring magnets can vibrate. If the mass distribution within the unit cell is kept constant while the damping effect is increased, then slightly wider stop bands can be obtained. In order to attain the maximum eddy current damping effect for the two different size ring magnets used in the analyses, an optimization study is performed. In this study, the copper tube mass and its inner radius are kept constant while its wall thickness and height are varied. Consequently, optimum geometry of the copper tube is determined. A prototype of the locally resonant periodic structure that includes eddy current damping is produced for experimental validation. During the assembly of the prototype, additional components are used within the unit cell for connection purposes. Their influence on magnetic flux density and eddy current density are observed via electromagnetic finite element analyses. It is seen that the stainless steel Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
16
components in the vicinity of the ring magnet and copper tube decrease the eddy current damping effect. For future work, non-magnetic materials can be used in order not to disturb the generated eddy currents in the copper tube. To sum up, eddy current damping is implemented to a one-dimensional locally resonant periodic structure. Moreover, analytical, numerical and experimental frequency response plots show matching stop bands and the same level of resonance peak amplitudes. Since this structure shows a vibration stop band in a desired frequency range with predetermined damping characteristics, it can be used as a vibration isolator or frequency filter. If the unit cell in this study is modified so that it becomes more slender, then it can be used as the building block of two- or three-dimensional locally resonant lattice structures with eddy current damping. Acknowledgement This work was supported by Bogazici University Research Fund with Grant Number 14A06P6.
References [1] J. Jensen, Phononic band gaps and vibrations in one and two dimensional mass-spring structures, J. Sound Vib. 266 (5) (2003) 1053–1078, http://dx.doi.org/ 10.1016/S0022-460X(02)01629-2. [2] A. Aly, A. Mehaney, H. Hanafey, Phononic band gaps in one dimensional mass spring system, in: PIERS Proceedings, vol. 1043, 2012, pp. 27–30. [3] G. Acar, C. Yilmaz, 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 (24) (2013) 6389–6404, http://dx.doi.org/10.1016/j.jsv.2013.06.022. [4] Y.-Z. Wang, F.-M. Li, K. Kishimoto, Y.-S. Wang, W.-H. Huang, Elastic wave band gaps in magnetoelectroelastic phononic crystals, Wave Motion 46 (1) (2009) 47–56. [5] Y.-Z. Wang, F.-M. Li, K. Kishimoto, Y.-S. Wang, W.-H. Huang, Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress, Eur. J. Mech.-A/Solids 29 (2) (2010) 182–189. [6] C. Yilmaz, G. Hulbert, N. Kikuchi, Phononic band gaps induced by inertial amplification in periodic media, Phys. Rev. B 76 (5) (2007) 1–9, http://dx.doi.org/ 10.1103/PhysRevB.76.054309. [7] C. Yilmaz, G. Hulbert, Theory of phononic gaps induced by inertial amplification in finite structures, Phys. Lett. A 374 (34) (2010) 3576–3584, http://dx.doi.org/ 10.1016/j.physleta.2010.07.001. [8] G. Wang, J. Wen, X. Wen, Quasi one dimensional phononic crystals studied using the improved lumped mass method application to locally resonant beams with flexural wave band gap, Phys. Rev. B 71 (10) (2005) 1–12, http://dx.doi.org/10.1103/PhysRevB.71.104302. [9] D. Yu, Y. Liu, G. Wang, H. Zhao, J. Qiu, Flexural vibration band gaps in timoshenko beams with locally resonant structures, J. Appl. Phys. 100 (12) (2006) 1–5, http: //dx.doi.org/10.1063/1.2400803. [10] Y. Liu, D. Yu, L. Li, H. Zhao, J. Wen, X. Wen, Design guidelines for flexural wave attenuation of slender beams with local resonators, Phys. Lett. A 362 (5) (2007) 344–347, http://dx.doi.org/10.1016/j.physleta.2006.10.056. [11] G. Wang, X. Wen, J. Wen, Y. Liu, Quasi one dimensional periodic structure with locally resonant band gap, J. Appl. Mech. 73 (1) (2006) 167–170, http://dx.doi.org/ 10.1115/1.2061947. [12] Y. Xiao, J. Wen, D. Yu, X. Wen, Flexural wave propagation in beams with periodically attached vibration absorbers: band-gap behavior and band formation mechanisms, J. Sound Vib. 332 (4) (2013) 867–893, http://dx.doi.org/10.1016/j.jsv.2012.09.035. [13] Y. Xiao, J. Wen, L. Huang, X. Wen, Analysis and experimental realization of locally resonant phononic plates carrying a periodic array of beam like resonators, J. Phys. D: Appl. Phys. 47 (4) (2014) 1–12, http://dx.doi.org/10.1088/0022-3727/47/4/045307. [14] A. Phani, M. Hussein, Dynamics of Lattice Materials, Wiley, U.S, 2016. [15] M.J. Frazier, M.I. Hussein, Viscous-to-viscoelastic transition in phononic crystal and metamaterial band structures, J. Acoust. Soc. Am. 138 (5) (2015) 3169–3180. [16] M.I. Hussein, M.J. Frazier, Metadamping: an emergent phenomenon in dissipative metamaterials, J. Sound Vib. 332 (20) (2013) 4767–4774. [17] M.J. Frazier, M.I. Hussein, Generalized bloch's theorem for viscous metamaterials: dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex, C. R. Phys. 17 (5) (2016) 565–577. [18] W. Saslow, Maxwells theory of eddy currents in thin conducting sheets, and applications to electromagnetic shielding and maglev, Am. J. Phys. 60 (8) (1992) 693–711, http://dx.doi.org/10.1119/1.17101. [19] C. MacLatchy, P. Backman, L. Bogan, A quantitative magnetic braking experiment, Am. J. Phys. 61 (12) (1993) 1096–1101, http://dx.doi.org/10.1119/1.17356. [20] K. Hahn, E. Johnson, A. Brokken, S. Baldwin, Eddy current damping of a magnet moving through a pipe, Am. J. Phys. 66 (12) (1998) 1066–1076, http://dx.doi.org/ 10.1119/1.19060. [21] Y. Levin, F. da Silveira, F.B. Rizzato, Electromagnetic braking: a simple quantitative model, Am. J. Phys. 74 (9) (2006) 815–817, http://dx.doi.org/10.1119/1.2203645. [22] S. Cheah, H.A. Sodano, Novel eddy current damping mechanism for passive magnetic bearings, J. Vib. Control 14 (11) (2008) 1749–1766, http://dx.doi.org/10.1177/ 1077546308091219. [23] J. Bae, J. Hwang, J. Park, D. Kwag, Modeling and experiments on eddy current damping caused by a permanent magnet in a conductive tube, J. Mech. Sci. Technol. 23 (11) (2009) 3024–3035, http://dx.doi.org/10.1007/s12206-009-0819-0. [24] G. Donoso, C. Ladera, P. Martin, Magnet fall inside a conductive pipe: motion and the role of the pipe wall thickness, Eur. J. Phys. 30 (4) (2009) 855–869, http://dx. doi.org/10.1088/0143-0807/30/4/018. [25] G. Donoso, C. Ladera, P. Martin, Damped fall of magnets inside a conducting pipe, Am. J. Phys. 79 (2) (2011) 193–200, http://dx.doi.org/10.1119/1.3531964. [26] M. Symans, M. Constantinou, Passive fluid viscous damping systems for seismic energy dissipation, ISET J. Earthq. Technol. 35 (4) (1998) 185–206. [27] H.A. Sodano, J. Bae, D. Inman, W. Belvin, Concept and model of eddy current damper for vibration suppression of a beam, J. Sound Vib. 288 (4) (2005) 1177–1196, http://dx.doi.org/10.1016/j.jsv.2005.01.016. [28] B. Ebrahimi, M. Khamesee, M. Golnaraghi, Design and modeling of a magnetic shock absorber based on eddy current damping effect, J. Sound Vib. 315 (4) (2008) 875–889, http://dx.doi.org/10.1016/j.jsv.2008.02.022. [29] C. Elbuken, M. Khamesee, M. Yavuz, Eddy current damping for magnetic levitation: downscaling from macro-to micro-levitation, J. Phys. D: Appl. Phys. 39 (18) (2006) 3932–3938, http://dx.doi.org/10.1088/0022-3727/39/18/002. [30] J. Bae, J. Hwang, J. Roh, J. Kim, M. Yi, J. Lim, Vibration suppression of a cantilever beam using magnetically tuned-mass-damper, J. Sound Vib. 331 (26) (2012) 5669–5684, http://dx.doi.org/10.1016/j.jsv.2012.07.020. [31] E. Pestel, F. Leckie, Matrix Methods in Elastomechanics, 435, McGraw-Hill, New York, 1963. [32] J.T. Broch, Mechanical vibration and shock measurements, Brüel & Kjær, 1980. [33] L. Brillouin, Wave propagation in periodic structures: electric filters and crystal lattices, Cour. Corp. (2003). [34] Y. Levin, F.L. da Silveira, F.B. Rizzato, Electromagnetic braking: a simple quantitative model, Am. J. Phys. 74 (9) (2006) 815–817, http://dx.doi.org/10.1119/1.2203645. [35] R. Singh, V. Singh, Analysis of eddy current damper for suppression of vibrations using comsol software, Eng. Solid Mech. 3 (4) (2015) 215–222, http://dx.doi.org/ 10.5267/j.esm.2015.7.004. [36] M.A. Salam, Electromagnetic Field Theories for Engineering, Springer Science & Business Media; Singapore, 2014. [37] D.J. Inman, R.C. Singh, Engineering Vibration, 3, Prentice Hall; New Jersey, 2001.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures Efe Ozkaya, Cetin Yilmaz n Department of Mechanical Engineering, Bogazici University, 34342, Bebek, Istanbul, Turkey
a r t i c l e i n f o
abstract
Article history: Received 11 April 2016 Received in revised form 31 October 2016 Accepted 16 November 2016 Handling Editor: A.V. Metrikine
The effect of eddy current damping on a novel locally resonant periodic structure is investigated. The frequency response characteristics are obtained by using a lumped parameter and a finite element model. In order to obtain wide band gaps at low frequencies, the periodic structure is optimized according to certain constraints, such as mass distribution in the unit cell, lower limit of the band gap, stiffness between the components in the unit cell, the size of magnets used for eddy current damping, and the number of unit cells in the periodic structure. Then, the locally resonant periodic structure with eddy current damping is manufactured and its experimental frequency response is obtained. The frequency response results obtained analytically, numerically and experimentally match quite well. The inclusion of eddy current damping to the periodic structure decreases amplitudes of resonance peaks without disturbing stop band width. & 2016 Elsevier Ltd All rights reserved.
Keywords: Phononic band gap Periodic structure Eddy current damping Local resonator
1. Introduction The frequency ranges, in which propagation of acoustic or elastic waves are hindered, are called phononic band gaps [1– 5]. Various methods can be used to generate phononic band gaps. In Bragg scattering method, the interference of the incident and reflected waves attenuates the energy of the propagating wave to create phononic band gaps [6,7]. In inertial amplification method, the wave propagating medium's effective inertia is amplified via embedded amplification mechanisms [3,6,7]. In local resonance method, when the frequency of the incident wave upon the periodic structure is close to the resonance frequency of the resonators, the wave energy excites the resonators and wave propagation is hindered. As a result, a band gap is formed around the resonance frequencies of the local resonators [8–13]. When there is no damping in a locally resonant structure, deep anti-resonance notches can be observed in its frequency response function enabling excellent vibration isolation around these frequencies [14]. However, in an undamped structure with local resonators, sharp resonance peaks are observed. Consequently, if the structure is excited around these frequencies, large undesired amplifications can be seen. Former studies have considered the effect of damping in lattices with local resonators [15–17]. Generally, viscous damping is investigated in these studies. In this paper, eddy current damping will be included to the local resonators in a periodic structure with the aim of decreasing the amplitudes of the resonance peaks while preserving a deep and wide stop band. Eddy currents are created when a non-magnetic conductor is moving in a region imposed upon a stationary magnetic field or when a time varying magnetic field has an impact upon a stationary non-magnetic conductor. According to Lenz's n
Corresponding author. E-mail addresses: [email protected] (E. Ozkaya), [email protected] (C. Yilmaz).
http://dx.doi.org/10.1016/j.jsv.2016.11.027 0022-460X/& 2016 Elsevier Ltd All rights reserved.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Law, eddy currents circulating in a conductor create their own magnetic field. As a result, a resistive force which opposes the motion of the moving part is created. When vibration between a conductor and a magnet is considered, a damping force proportional to the velocity of the vibration is generated and this is called eddy current damping [18–25]. Both viscous and eddy current damping forces are proportional to the velocity of the vibration. Therefore, mathematically these two damping mechanisms are equivalent. However, when the physical structure of these dampers are considered, one can see that eddy current damping offers certain advantages. In viscous dampers, generally a fluid is forced through a small opening by a moving piston. Due to sliding motion, there may be wear problems. Moreover, compressibility of the fluid inside these dampers introduce some stiffness besides damping [26]. Stiffness change due to compressibility effect is more pronounced at high frequencies. If eddy current damping is added to a structure, its stiffness is not disturbed as eddy current damping is a non-contact type of damping. Due to its non-contact nature, it is not prone to wear. Eddy current damping method has been used in various applications. Sodano et al. [27] used eddy current damping to impede the transverse vibrations of a cantilever beam. Ebrahimi et al. [28] studied a design that involves two permanent magnets and a conductive aluminum plate to generate both stiffness and variable damping. Elbuken et al. [29] applied eddy current damping to magnetically levitated miniaturized objects. Bae et al. [30] introduced a tuned mass damper with eddy current damping. In this paper, eddy current damping will be combined with the local resonance method for the first time. In order to generate eddy current damping, concentric copper tube and magnet assemblies will be used. To enforce one-dimensional (1D) motion between these components, a novel periodic structure with parallel spiral springs will be designed. Parametric studies will be conducted with the aim of obtaining a wide local resonance induced gap. The periodic structure will be manufactured to see the effect of eddy current damping on resonance peaks, stop band width, and depth in frequency response plots.
2. Models and methodology The study consists of three main sections. Firstly, analytical calculations are done to determine the values of the design parameters. Depending on these parameter values, a prototype design is established numerically and modal analysis is conducted. In the final phase, experimental validation is performed for the prototype produced. In this study, small deformations are considered, hence the structure is assumed to be linear and strength is not an issue. 2.1. Analytical model of the locally resonant periodic structure A one-dimensional locally resonant periodic structure that includes eddy current damping is modelled as in Fig. 1. During modelling of the periodic structure, five parameters are taken into account, which are resonator mass (mr), total mass of the remaining parts in the unit cell when the resonator mass is excluded (m), stiffness of the spring connecting the resonator mass to the unit cell (kr), total stiffness between the unit cells (k) and the eddy current damping constant (c). As eddy current damping force is proportional to the velocity of vibration, c can be considered as a viscous damping constant. Phononic band structure and the frequency response of the periodic structure will be determined by the transfer matrix method [31]. Through mechanical impedance to mass conversion, periodic structure's effective mass can be calculated to be used in its overall transfer matrix. Mechanical impedance representations for mass, spring and damper elements are jωm, k/jω and c, respectively [32]. By using these impedances, effective mass of the unit cell of the system shown in Fig. 1(b) is obtained as follows:
meff = m +
1 1 ω2 − mr k r + jωc
(1)
The point transfer matrix for meff (Eq. (2)) and the field transfer matrix for k (Eq. (3)) are used [31] to obtain the relationship between consecutive unit cells (Eq. (5)) through the state vector relationship as shown in Eq. (4).
Fig. 1. Lumped parameter model of the locally resonant periodic structure including eddy current damping (a) 1D array with local resonators, (b) the equivalent array with effective mass.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
⎡ 1 0⎤ ⎥ Pi = ⎢ 2 ⎣ − meff ω 1⎦
(2)
⎡ 1⎤ ⎢1 ⎥ Fi = ⎢ k⎥ ⎣ 0 1⎦
(3)
⎡ u ⎤R ziR = ⎢ ⎥ ⎣ N ⎦i
(4)
ziR = Pi Fi ziR− 1
(5)
where u is displacement and N is internal force. To construct the band structure of the locally resonant periodic structure that includes eddy current damping, Bloch's Theorem (Eq. (6)) is applied [33].
ziR = ziR− 1e jγ
(6)
Here, γ represents the wave number. Eqs. (4), (5) and (6) can be used to obtain the relationship for a unit cell with periodic boundary conditions (Eq. (7)).
⎡ 1⎤ R 1 0⎤ ⎢ 1 ⎡ u ⎤R ⎡ ⎥⎡ u ⎤ k ⎥ ⎢⎣ ⎥⎦ e−jγ ⎢⎣ N ⎥⎦ = ⎢ − m ω2 1⎥ ⎢ N ⎣ ⎦ eff i i ⎣0 1⎦
(7)
Due to periodic boundary conditions, the determinant expressed in Eq. (8) is obtained.
1 − e jγ
1 k
− meff ω2
−meff ω2 + 1 − e jγ k
=0 (8)
Consequently, dispersion equation (Eq. (9)) for the periodic structure including infinite number of unit cells is attained. ω values that satisfy this dispersion equation are obtained for γ values in between 0 and π. As a result, the five aforementioned design parameters can be optimized for a desired band gap width.
1 − 2e jγ + e2jγ +
meff ω2e jγ =0 k
(9)
On the other hand, to obtain the transmissibility plot for the periodic structure with n unit cells, overall transfer matrix of the structure should be calculated by using Eqs. (2) and (3). In the overall transfer matrix, the U22 entry is used to find the displacement or acceleration transmissibility (TR (ω))(see Eq. (11))
⎡ U11 U12 ⎤ Uoverall = (Pi Fi )n = ⎢ ⎥ ⎣ U21 U22 ⎦ TR (ω) =
n = 1, 2, 3, …
x (ω) x¨ (ω) 1 = = y (ω) y¨ (ω) U22
(10)
(11)
where x (ω) and y (ω) are the output and input displacements. Moreover, the frequency range where TR (ω) < 1 is defined as the stop band. In order to show the effect of eddy current damping on the frequency response of the periodic structure, its lumped parameter model is optimized to obtain a stop band between 50 and 100 Hz frequency range. During optimization, parametric studies are conducted by using the dispersion equation (Eq. (9)) and the transmissibility (Eq. (11)) of the lumped parameter model shown in Fig. 1. 2.2. Finite element models of the periodic structure Eddy current damping within the periodic structure will be realized through ring magnet and copper tube assemblies. The magnets will be constrained to move concentrically inside the copper tubes with small amplitudes. Hence, orientation of the magnetic field lines generated by each magnet or magnetic flux density within the corresponding copper tube will change negligibly during vibration. Consequently, the damping coefficient generated by each magnet can be assumed as constant for small amplitude vibrations. Electromagnetic finite element models of these assemblies will be formed to calculate their damping coefficients. Moreover, structural finite element models of the periodic structure will be generated Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Fig. 2. Band structure plots for three different mass ratios (a) m/mr = 1/2, (b) m/mr = 1/3, (c) m/mr = 1/4 . In all plots m = 0.1 kg , k ¼ 126 kN/m, c ¼0 N s/m and kr /mr = 2π × 50 rad/s.
to be used for modal analysis. The equivalent viscous damping coefficients of the magnet and copper tube assemblies will be used in the structural finite element and lumped parameter models. Finally, frequency response results of the lumped parameter and finite element models will be compared.
3. Parametric studies First two parametric studies focuses on the effect of mass distribution within the unit cell and the number of unit cells to the width and depth of the stop band of the periodic structure. While the latter two focuses on the effect of eddy current damping to the frequency response of the periodic structure. First of all, the effect of remaining mass of the unit cell when the resonator is excluded (m) to resonator mass (mr) ratio is investigated. In this study, two different ring magnets with masses 0.032 kg and 0.052 kg are planned to be used for eddy current damping purposes. Thus, the remaining mass (m) of the unit cell including these magnets is predicted to be around 0.1 kg . On the other hand, the resonator mass (mr) which mainly consists of copper tube can be varied to change the band gap frequency range. Fig. 2 shows that as m /mr ratio decreases, wider band gaps can be obtained. It is seen that m /mr = 1/3 provides a band gap in the target frequency range (50–100 Hz). Secondly, the effect of number of unit cells to the width and depth of the stop band is investigated. Fig. 3 shows that as the number of unit cells is increased, anti-resonance depth at 50 Hz also increases. Therefore, vibration isolation performance in between 50 and 100 Hz is improved. However, as the number of unit cells increases, the overall size and mass of
Fig. 3. The effect of number of unit cells in the undamped periodic structure on the depth of the stop band between 50 and 100 Hz. Here, m = 0.1 kg , mr = 0.3 kg , k ¼126 kN/m, kr ¼ 29.6 kN/m.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Fig. 4. The effect of damping ratio of the resonators on stop band width, resonance peaks and anti-resonance depth for a periodic structure with four unit cells. Here, m = 0.1 kg , mr = 0.3 kg , k ¼126 kN/m, kr ¼ 29.6 kN/m.
the periodic structure increases, as well. Considering the compromise between stop band depth and mass of the periodic structure, four unit cells are chosen to be used in the design. Within the unit cell of the periodic structure, a ring magnet will be used inside a copper tube to generate eddy current damping. In the third parametric study, the effect of damping ratio on the stop band width and depth of the structure will be investigated. Only the eddy current damping constant or equivalent viscous damping coefficient (c) will be varied in the lumped parameter model of the periodic structure. Fig. 4 shows that as the damping ratio of the resonators (see Eq. (12)) increases, all the resonance peaks and the anti-resonance depth at 50 Hz decreases.
ζ=
c 2 mr k r
(12)
Also note that the increase in the damping ratio slightly increases the stop band width. However, if larger damping coefficients are to be realized by the use of larger magnets in the design, then the mass of the unit cell excluding the resonator (m) will increase. Consequently, m /mr ratio will increase, resulting in a narrower stop band as shown in Fig. 2. In the literature, many parametric studies are conducted regarding cylindrical magnets moving inside copper tubes. It has been shown that eddy current damping force or Lorentz force is affected by the cylindrical magnet's radius, height, and grade; copper tube's height, wall thickness and inner diameter (or the air gap between the tube and the magnet) [19,20,24,25,34,35]. Similarly, eddy current damping force or Lorentz force generated by a ring magnet can be affected by the magnet dimensions and grade, and copper tube dimensions. Two set of NdFeB ring magnets (see Table 1) will be used in the locally resonant periodic structure to generate eddy current damping. Notice that the remanent flux densities of these N35 grade magnets can be in a quite wide range depending on the manufacturer. In order to determine the remanent flux densities of these magnets accurately, free fall experiments are conducted. To that end, a 550 mm long 1.5 mm thick copper tube with 32 mm inner diameter is used in which the ring magnets are let to free fall. The free fall times of the ring magnets are measured with a digital chronometer. The average of 10 measurements are recorded. In line with the literature [19,20,24,25,34,35], the acceleration times of the magnets are neglected and the magnets are assumed to descent with terminal velocity during their entire fall. Later, the length of the copper tube is divided by these free fall times and terminal velocities are obtained. Since at terminal velocity (vterminal) Lorentz force (Florentz) is equal to the weight of the magnet (mg), by dividing the Lorentz force by the terminal velocity, damping constant (c) can be obtained as in Eq. (13).
c=
Florentz mg = vterminal vterminal
(13)
Following this experiment, an electromagnetic finite element model which is a duplicate of this experiment is prepared. During the finite element analysis, remanent flux densities of each ring magnet are varied from 0.9 T to 1.5 T with 0.01 T increments to get the same terminal velocity values as in the free fall experiment. In Table 2, corresponding remanent flux Table 1 Dimensions, weights and remanent flux density intervals of the ring magnets.
Magnet outer radius (mm ) Magnet inner radius (mm ) Magnet height (mm ) Magnet weight (N ) Remanent flux density interval (T)
Small magnet
Large magnet
12.5 4.25 10 0.31 0.9–1.5
15.35 4.25 10 0.51 0.9–1.5
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Table 2 Free fall experiment results for ring magnets. Here, the copper tube is 550 mm long, 1.5 mm thick and its inner diameter is 32 mm.
Measured free fall time (s) Terminal velocity (m/s) Lorentz force (N) Damping constant (N s/m) Remanent flux density (T)
Small magnet
Large magnet
3.18 0.173 0.31 1.81 1.03
9.59 0.057 0.51 8.90 1.14
Fig. 5. Computational analysis for the free fall of the large ring magnet inside a copper tube with 11 mm wall thickness and 23.2 mm height. (a) Section view, (b) 2D axi-symmetric meshed geometry, (c) magnetic flux density norm (T) contour plot, (d) current density norm (A/m2 ) contour plot, (e) relative velocity between magnet and copper tube, and (f) Lorentz force between magnet and copper tube that is generated due to eddy currents within the copper tube.
densities are given. A final parametric study will be conducted to obtain the maximum amount of damping for the given ring magnets. To that end, an electromagnetic finite element model of the ring magnets and copper tube will be generated. The magnets will Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
Fig. 6. The effect of copper tube's wall thickness on damping behaviour for a constant resonator mass of 0.3 kg . (a) Effect of copper the wall thickness on damping constant, (b) effect of copper tube wall thickness on damping ratio. Here, mr = 0.3 kg , kr ¼29.6 kN/m.
be subjected to free fall due to gravity and the Lorentz force between magnets and copper tubes will be calculated. During the analyses, mass of the copper tube is kept constant at 0.3 kg . Moreover, its inner diameter is fixed at 32 mm while its height and wall thickness is varied considering the mass constraint. Height of the tube is decreased as the wall thickness is increased from 1 mm to 20 mm with 1 mm increments. One of these analyses is shown in Fig. 5. Here, the wall thickness of the copper ( ρ = 8700 kg/m3) tube is 11 mm, and its height is 23.2 mm satisfying the mass constraint of 0.3 kg . Notice that as the magnet velocity reaches a terminal value, the Lorentz force in the direction of free fall (z-direction) reaches 0.51 N, which is the weight of the large magnet. Then, damping constant can be found using Eq. (13). For this case, vterminal = 0.0167 m/s giving c¼30.5 N.s/m. Fig. 6 shows the effect of wall thickness of the copper tube on the eddy current damping constant and the damping ratio of the resonator. It is seen that to acquire the maximum damping ratio under a mass constraint, there is an optimum copper wall thickness value. For a copper tube having an inner diameter of 32 mm with a mass constraint of 0.3 kg , the optimum wall thickness value is around 11 mm.
4. Design and finite element analysis of the locally resonant periodic structure In Fig. 7, lumped parameter model of the unit cell of the periodic structure and the corresponding distributed parameter model can be seen. In Fig. 7(b) two sets of parallel spiral springs are used to realize the stiffnesses k and kr. The two parallel spiral springs (k r /2) that are attached to the copper tube (mr) allow the ring magnet to move concentrically inside the tube without any contact. Notice that as the large magnet diameter is 30.7 mm and the inner diameter of the tube is 32 mm, there is very little clearance between them. The other two parallel spiral springs (k/2) are used to connect the unit cells with each other. Due to their parallel design, they mainly allow axial relative motion between the unit cells. As a result, a novel onedimensional locally resonant periodic structure can be realized with prescribed stiffness characteristics for axial excitations. In the analyses with the lumped parameter model, the stiffnesses kr and k are determined as 29.6 kN/m and 126 kN/m (see Figs. 3 and 4). In order to realize these stiffness values, the spiral springs shown in Fig. 8(a) and 8(b) are designed. Both of these stainless steel (E ¼193 GPa, ρ = 7750 kg/m3, ν = 0.31) springs have 60 mm outer diameter, 8.5 mm inner diameter and 0.7 mm thickness. The spiral shaped slots are chosen such that the stiffness of these springs are close to half of the stiffness values in the lumped parameter model. The locally resonant periodic structure with four unit cells can be seen in Fig. 9. The whole structure has an outer diameter of 60 mm and length of 376 mm. Notice that the central stainless steel M8 bolts connect all parts within each unit cell. The bolt head is connected to a stainless steel circular plate that has 1 mm thickness. Moreover, three stainless steel M4
Fig. 7. (a) Lumped parameter model of the unit cell. (b) Distributed parameter model of the unit cell.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
Fig. 8. (a) Spiral spring with kr /2 = 13.5 kN/m stiffness. (b) Spiral spring with k/2 = 62.0 kN/m stiffness.
Fig. 9. The locally resonant periodic structure with four unit cells. (a) Isometric view, (b) side view.
bolts are used near the periphery of each circular plate to make the connection with the neighbouring unit cell. Small plastic rings and nuts with negligible masses will be used during the assembly of the locally resonant periodic structure. However, as the effect of these plastic parts are assumed to be negligible, they are not included in the model shown in Fig. 9. In the parametric studies with the lumped parameter model, m and mr were taken as round numbers like 100 g and 300 g. When the distributed parameter model of the structure is formed, m is calculated as 0.143 kg for the unit cell with small magnet. However, mr is 0.392 kg giving m /mr = 0.365 which is close to the target mass ratio of 1/3. Furthermore, m becomes 0.162 kg and 0.111 kg for the unit cell with large magnet and no magnet. As a result, mass ratio for the large magnet and no magnet cases are 0.413 and 0.283, respectively. The copper tube with mass mr in Fig. 7(b) does not have constant wall thickness as in Fig. 5(a). Notice that the wall thickness is smaller on the two faces allowing room for out-of-plane deformations of the spiral springs (k r /2). The wall thickness in the middle cross-section is chosen as 14 mm, which is a little higher than the optimum value of 11 mm to compensate the decrease in the wall thickness on the two faces. In Fig. 10, electromagnetic finite element analysis of the designed copper tube shape can be seen. Here, the large magnet is shown inside the tube. The same analysis is conducted for the small magnet, as well. As a result of these analyses, the damping constants for the small and large magnets are found as 4.02 N.s/m and 20.4 N.s/m, respectively. These values are a little lower than the peak values in Fig. 6(a). When the eddy current density plots in Fig. 5(d) and Fig. 10(d) are compared, higher current densities are seen over larger areas in Fig. 5(d). This explains the lower damping constants for the designed copper tube. The relationship between magnetic flux density (B) and magnetic field (H) is given as
B = μH
(14)
where μ represents the permeability of the medium [36]. Therefore, a change in magnetic field results in a change in magnetic flux density as μ is constant. If magnetic materials like the stainless steel parts in the resonator are present near a magnet, then they can distort the magnetic field lines. As a result, magnetic flux density can change. In order to check the effect of other elements within the resonator on the damping behaviour, a more detailed model of the resonator is analysed. In Fig. 11, the stainless steel (σ = 1.32 × 106 Siemens/m ) spiral springs k r /2, stainless steel M8 bolt in the middle and the plastic rings that are used to separate the magnet and the spiral springs are included in the model. As a result of this analysis, the damping constants for the small and large magnets are found to be 2.31 N.s/m and 10.8 N. Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
9
Fig. 10. (a) Section view of a large ring magnet inside the designed copper tube. (b) 2D axi-symmetric finite element model of the designed copper tube and large ring magnet. (c) Magnetic flux density norm (T) contour plot. (d) Current density norm (A/m2) contour plot.
Fig. 11. (a) Section view of the fully represented resonator. (b) Fully represented 2D axi-symmetric resonator geometry with large magnet included. (c) Magnetic flux density norm (T) contour plot. (d) Current density norm (A/m2) contour plot.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
10
Fig. 12. Finite element mesh for the locally resonant periodic structure with small magnets included. The resonator of the second unit cell is suppressed for visual purposes.
s/m, respectively. The smaller damping constant values in this analysis can be explained by observing that the stainless steel bolt and stainless steel spiral springs reduce magnetic flux density close to the copper tube (compare Figs. 10(c) and 11(c)). Consequently, smaller eddy current densities are obtained in Fig. 11(d) when compared to Fig. 10(d). Notice that the interaction between magnets is not considered in this study since the magnets are far away from each other as shown in Fig. 9. However, if the unit cell length was smaller, then each magnet would interact with the neighbouring magnets which in turn would change the stiffness characteristics of the unit cell. In that case, the magnetic field lines of each magnet would also interact with the neighbouring copper tubes that would change the damping characteristics, as well.
4.1. Structural finite element analyses of the locally resonant periodic structure The finite element model of the locally resonant periodic structure shown in Fig. 9 is formed and its first 10 natural frequencies are determined for three cases: resonators with no magnets, resonators with small magnets, and resonators with large magnets. The finite element mesh for the case with small magnets can be seen in Fig. 12. Similar mesh sizes are used in the other two cases. In all cases, the structures are fixed from the right side and frictionless supports are used around the resonators. Table 3 shows the first 10 natural frequencies of the locally resonant periodic structure without any magnets. When the corresponding mode shapes are investigated, it is seen that the 5th mode is torsional as shown in Fig. 13(b). Therefore, this mode is not expected to be seen in the frequency response if the system is excited axially. As a result, the stop band will occur between the 4th and 6th modes, which are shown in Figs. 13(a) and 13(c). Second modal analysis is done when small ring magnets are inserted inside the copper tubes. The first 10 natural frequencies are shown in Table 4. The stop band will occur in between 4th and 6th modes and they are depicted in Fig. 14 with the torsional mode in between at 62.9 Hz. These limiting mode shapes are shown in Figs. 14(a) and 14(c). Finally, a modal analysis is done when large ring magnets are inserted inside the copper tubes. The first 10 natural frequencies are shown in Table 5. The stop band will again occur in between 4th and 6th modes with a torsional mode in between and they are shown in Fig. 15.
5. Experimental measurements and results 5.1. Laser vibrometer measurements to estimate the eddy current damping constants To validate the damping constant values obtained through the electromagnetic finite element analysis, measurements are taken via a laser vibrometer for the three types of resonators. Fig. 16 shows one of these resonators. In the first measurement no ring magnet is inserted inside the copper tube. In the second and third measurements, small and large ring magnets are inserted inside the copper tube to create eddy current damping. The damping coefficient in all three case are calculated through Eq. (15).
c = 2ζ meff k r
(15)
In the experiments, the copper tube is fixed and the inner part is allowed to vibrate. Thus, effective mass (meff) used in Eq. (15) differs depending on the mass of the magnet used inside the resonator (see Table 6). In Eq. (15), kr ¼27 kN/m and ζ is determined by the logarithmic decrement method. Generally, this method can be formulated as follows [37]: Table 3 First 10 natural frequencies of the periodic structure without any magnets. In the last row, types of the modes are indicated as axial (A) or torsional (T). Mode number Frequency (Hz) Mode type
1 30.7 A
2 45.2 A
3 47.2 A
4 47.7 A
5 66.2 T
6 130.8 A
7 170.5 T
8 231.5 T
9 257.6 A
10 258.3 T
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
11
Fig. 13. Mode shapes of periodic structure without any ring magnets that determine the lower and upper limits of the stop band with a torsional mode in between. (a) 4th mode shape at 47.7 Hz , (b) 5th mode shape at 66.2 Hz and (c) 6th mode shape at 130.8 Hz . Table 4 First 10 natural frequencies of the periodic structure including small ring magnets. In the last row, types of the modes are indicated as axial (A) or torsional (T). Mode number Frequency (Hz) Mode type
δ=
ζ=
1 28.7 A
2 43.1 A
3 44.6 A
1 ⎛ x (t ) ⎞ ln ⎜ ⎟ n ⎝ x (t + nT ) ⎠
4 46.5 A
5 62.9 T
6 111.3 A
7 163.2 T
8 215.4 T
9 220.1 A
10 244.3 T
(16)
δ 4π 2
+ δ2
(17)
where δ is the logarithmic decrement, x(t) is the peak displacement amplitude at time t, x (t + nT ) is the peak displacement amplitude after n periods (T). Three different measurement results are shown in Fig. 17 with six data points taken in order to apply the logarithmic decrement method given in Eqs. (16) and (17). Notice that in Eq. (16) displacement amplitude information is used. However, vibration velocity is measured by the laser vibrometer. Therefore, Eq. (16) needs to be modified. Since this equation is derived by assuming that the motion is underdamped, displacement amplitudes are related by Eq. (18)
x (t + nT ) = e−ζω n nT x (t )
(18)
where ωn is the undamped natural frequency of the system. When we take the time derivative of Eq. (18) the following equation is obtained:
v (t + nT ) = e−ζω n nT v (t )
(19)
where v(t) is the velocity amplitude at time t, v (t + nT ) is the velocity amplitude after n periods (T). There is a phase difference between peak velocity and peak displacement values. Nevertheless, for all values of t, the following relationship holds:
x (t )/x (t + nT ) = v (t )/v (t + nT ) = e ζω n nT
(20)
Therefore, Eq. (20) can be used for the peak values in the velocity measurements. As a result, Eq. (16) can be reformulated as Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
12
Fig. 14. Mode shapes of periodic structure with small magnets that determine the lower and upper limits of the stop band with a torsional mode in between. (a) 4th mode shape at 46.5 Hz , (b) 5th mode shape at 62.9 Hz and (c) 6th mode shape at 111.3 Hz . Table 5 First 10 natural frequencies of the periodic structure including large magnets. In the last row, types of the modes are indicated as axial (A) or torsional (T). Mode number Frequency (Hz) Mode type
1 28.9 A
2 43.0 A
3 45.1 A
4 46.7 A
5 64.5 T
6 106.1 A
7 164.3 T
8 202.5 T
9 221.8 A
10 245.5 T
Fig. 15. Mode shapes of periodic structure with large magnets that determine the lower and upper limits of the stop band with a torsional mode in between. (a) 4th mode shape at 46.7 Hz , (b) 5th mode shape at 64.5 Hz and (c) 6th mode shape at 106.1 Hz .
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
13
Fig. 16. Measured resonator with a small ring magnet inside.
Table 6 Calculated damping ratios and constants via logarithmic decrement method.
Time (s) Velocity (m/s) δ ζ meff (kg) kr (kN/m) c (N s/m)
No magnet
Small magnet
Large magnet
7th peak
15th peak
7th peak
14th peak
3rd peak
6th peak
0.0423 0.0396 0.0387 0.0062 0.034 27.0 0.37
0.0958 0.0291
0.0544 0.0142 0.1537 0.0245 0.066 27.0 2.06
0.1158 0.00485
0.0225 0.00716 0.6882 0.1088 0.085 27.0 10.43
0.0529 0.000909
Fig. 17. Eddy current damping effect measured with laser vibrometer.
δ=
1 ⎛ v (t ) ⎞ ln ⎜ ⎟ n ⎝ v (t + nT ) ⎠
(21)
In Table 6, peak points taken for each measurement to be used in logarithmic decrement method are given, which are followed by the calculated damping ratios and damping constants. If the damping constant of the no magnet case is subtracted from the other cases, eddy current damping constants for each ring magnet can be obtained. Notice that these damping constant values are close to the values obtained through the fully represented resonator model (Fig. 11). Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
14
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 18. Experimental setup.
5.2. Experimental modal analysis of the prototype The prototype of the locally resonant periodic structure and the experimental setup can be seen in Fig. 18. Notice that the prototype is hanged by rubber cords to simulate free boundary conditions. Excitation is given from one end by a modal shaker. The input acceleration is measured through an impedance head and the output acceleration on the other end of the prototype is measured by an accelerometer. Transmissibility is calculated as the ratio of output acceleration to input acceleration (see Eq. (11)). In the three measurements taken, modal shaker excites the structure up to 400 Hz. In the first measurement, to observe the effect of structural damping within the prototype on the vibration stop band, ring magnets are not inserted inside the copper tubes. During the second and third measurements, small and large ring magnets are used, respectively. The effect of eddy current damping can be seen in Fig. 19 via the decrease in all resonance peaks. The stop band width does not change much due to magnet usage. For comparison purposes, frequency response plots are obtained analytically and numerically using the lumped parameter and finite element models, respectively. In Fig. 20, frequency response plots for the lumped parameter models can be seen. From the analytical calculations done by using the lumped parameter model without ring magnets, stop band is obtained in between 40.1 and 88.8 Hz for mr = 0.392 kg , m = 0.111 kg , k¼ 124 kN/m and kr ¼27 kN/m. When small ring magnets are included, stop band is obtained in between 38.7 and 80.9 Hz for mr = 0.392 kg , m = 0.143 kg , k ¼124 kN/m and kr ¼27 kN/m. It is seen that only the m value changes due to small ring magnet inclusion. This causes a decrease in stop band width due to mass ratio (m /mr ) increase. Finally, when the large ring magnets are inserted inside the copper tubes, stop band is obtained in between 37.4 and 78.8 Hz for mr = 0.392 kg , m = 0.162 kg , k¼ 124 kN/m and kr ¼27 kN/m. Inclusion of large ring magnets with higher damping decreases the stop band width. On the other hand, the resonance peaks are suppressed the most with the use of large ring magnets. Hence, there is a compromise between stop band width and damping. Fig. 21 shows the frequency response plots for the finite element models. Notice that the frequency response plots obtained analytically, numerically and experimentally are quite similar (see Figs. 19–21). There are small peaks around 60 Hz and 160 Hz in the numerical frequency response plot due to the torsional modes (see Tables 3–5). However, these peaks are insignificant and do not change the stop band characteristics. In Table 7, lower and upper stop band limits, and their ratios are presented. One can see that there are some shifts in the stop band frequency limits obtained through three different approaches. Nevertheless, ωu/ωl ratios are almost the same.
Fig. 19. Frequency response plots obtained from the experimental measurements.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
15
Fig. 20. Frequency response plots obtained via the lumped parameter models.
Fig. 21. Frequency response plots obtained via the finite element models.
Table 7 Lower (ωl) and upper (ωu) stop band limits, and their ratios (ωu/ωl ) obtained from the analytical, numerical and experimental FRF plots. Here, all frequencies are in Hz. No magnet
Analytical Numerical Experimental
Small magnet
Large magnet
ωl
ωu
ωu/ωl
ωl
ωu
ωu/ωl
ωl
ωu
ωu/ωl
40.1 47.7 52
88.8 105.2 110
2.21 2.20 2.11
38.7 43.6 46.1
80.9 92 100.4
2.09 2.11 2.18
37.4 38.2 39.1
78.8 89.3 100.4
2.11 2.34 2.57
6. Conclusions In this paper, a novel one-dimensional locally resonant periodic structure is designed to generate a stop band at low frequencies. In order to analyse the periodic structure, a lumped parameter and a finite element model are generated. Within the unit cell of the periodic structure, two sets of spiral springs are used. Axial relative motion between the unit cells is accomplished by the parallel placement of these spiral springs. Parametric studies regarding the effect of mass distribution within the unit cell and the number of unit cells on the stop band characteristics are conducted. It is seen that when the unit cell mass excluding the resonator to resonator mass ratio is about one-third, the targeted stop band width is achieved. As the number of unit cells increases, deeper stop bands are obtained with the burden of increasing the overall size and mass of the periodic structure. Considering the compromise between stop band depth and mass of the periodic structure, four unit cells are chosen. Furthermore, eddy current damping effect is included to the periodic structure through resonator masses being made of copper tubes in which ring magnets can vibrate. If the mass distribution within the unit cell is kept constant while the damping effect is increased, then slightly wider stop bands can be obtained. In order to attain the maximum eddy current damping effect for the two different size ring magnets used in the analyses, an optimization study is performed. In this study, the copper tube mass and its inner radius are kept constant while its wall thickness and height are varied. Consequently, optimum geometry of the copper tube is determined. A prototype of the locally resonant periodic structure that includes eddy current damping is produced for experimental validation. During the assembly of the prototype, additional components are used within the unit cell for connection purposes. Their influence on magnetic flux density and eddy current density are observed via electromagnetic finite element analyses. It is seen that the stainless steel Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i
E. Ozkaya, C. Yilmaz / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
16
components in the vicinity of the ring magnet and copper tube decrease the eddy current damping effect. For future work, non-magnetic materials can be used in order not to disturb the generated eddy currents in the copper tube. To sum up, eddy current damping is implemented to a one-dimensional locally resonant periodic structure. Moreover, analytical, numerical and experimental frequency response plots show matching stop bands and the same level of resonance peak amplitudes. Since this structure shows a vibration stop band in a desired frequency range with predetermined damping characteristics, it can be used as a vibration isolator or frequency filter. If the unit cell in this study is modified so that it becomes more slender, then it can be used as the building block of two- or three-dimensional locally resonant lattice structures with eddy current damping. Acknowledgement This work was supported by Bogazici University Research Fund with Grant Number 14A06P6.
References [1] J. Jensen, Phononic band gaps and vibrations in one and two dimensional mass-spring structures, J. Sound Vib. 266 (5) (2003) 1053–1078, http://dx.doi.org/ 10.1016/S0022-460X(02)01629-2. [2] A. Aly, A. Mehaney, H. Hanafey, Phononic band gaps in one dimensional mass spring system, in: PIERS Proceedings, vol. 1043, 2012, pp. 27–30. [3] G. Acar, C. Yilmaz, 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 (24) (2013) 6389–6404, http://dx.doi.org/10.1016/j.jsv.2013.06.022. [4] Y.-Z. Wang, F.-M. Li, K. Kishimoto, Y.-S. Wang, W.-H. Huang, Elastic wave band gaps in magnetoelectroelastic phononic crystals, Wave Motion 46 (1) (2009) 47–56. [5] Y.-Z. Wang, F.-M. Li, K. Kishimoto, Y.-S. Wang, W.-H. Huang, Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress, Eur. J. Mech.-A/Solids 29 (2) (2010) 182–189. [6] C. Yilmaz, G. Hulbert, N. Kikuchi, Phononic band gaps induced by inertial amplification in periodic media, Phys. Rev. B 76 (5) (2007) 1–9, http://dx.doi.org/ 10.1103/PhysRevB.76.054309. [7] C. Yilmaz, G. Hulbert, Theory of phononic gaps induced by inertial amplification in finite structures, Phys. Lett. A 374 (34) (2010) 3576–3584, http://dx.doi.org/ 10.1016/j.physleta.2010.07.001. [8] G. Wang, J. Wen, X. Wen, Quasi one dimensional phononic crystals studied using the improved lumped mass method application to locally resonant beams with flexural wave band gap, Phys. Rev. B 71 (10) (2005) 1–12, http://dx.doi.org/10.1103/PhysRevB.71.104302. [9] D. Yu, Y. Liu, G. Wang, H. Zhao, J. Qiu, Flexural vibration band gaps in timoshenko beams with locally resonant structures, J. Appl. Phys. 100 (12) (2006) 1–5, http: //dx.doi.org/10.1063/1.2400803. [10] Y. Liu, D. Yu, L. Li, H. Zhao, J. Wen, X. Wen, Design guidelines for flexural wave attenuation of slender beams with local resonators, Phys. Lett. A 362 (5) (2007) 344–347, http://dx.doi.org/10.1016/j.physleta.2006.10.056. [11] G. Wang, X. Wen, J. Wen, Y. Liu, Quasi one dimensional periodic structure with locally resonant band gap, J. Appl. Mech. 73 (1) (2006) 167–170, http://dx.doi.org/ 10.1115/1.2061947. [12] Y. Xiao, J. Wen, D. Yu, X. Wen, Flexural wave propagation in beams with periodically attached vibration absorbers: band-gap behavior and band formation mechanisms, J. Sound Vib. 332 (4) (2013) 867–893, http://dx.doi.org/10.1016/j.jsv.2012.09.035. [13] Y. Xiao, J. Wen, L. Huang, X. Wen, Analysis and experimental realization of locally resonant phononic plates carrying a periodic array of beam like resonators, J. Phys. D: Appl. Phys. 47 (4) (2014) 1–12, http://dx.doi.org/10.1088/0022-3727/47/4/045307. [14] A. Phani, M. Hussein, Dynamics of Lattice Materials, Wiley, U.S, 2016. [15] M.J. Frazier, M.I. Hussein, Viscous-to-viscoelastic transition in phononic crystal and metamaterial band structures, J. Acoust. Soc. Am. 138 (5) (2015) 3169–3180. [16] M.I. Hussein, M.J. Frazier, Metadamping: an emergent phenomenon in dissipative metamaterials, J. Sound Vib. 332 (20) (2013) 4767–4774. [17] M.J. Frazier, M.I. Hussein, Generalized bloch's theorem for viscous metamaterials: dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex, C. R. Phys. 17 (5) (2016) 565–577. [18] W. Saslow, Maxwells theory of eddy currents in thin conducting sheets, and applications to electromagnetic shielding and maglev, Am. J. Phys. 60 (8) (1992) 693–711, http://dx.doi.org/10.1119/1.17101. [19] C. MacLatchy, P. Backman, L. Bogan, A quantitative magnetic braking experiment, Am. J. Phys. 61 (12) (1993) 1096–1101, http://dx.doi.org/10.1119/1.17356. [20] K. Hahn, E. Johnson, A. Brokken, S. Baldwin, Eddy current damping of a magnet moving through a pipe, Am. J. Phys. 66 (12) (1998) 1066–1076, http://dx.doi.org/ 10.1119/1.19060. [21] Y. Levin, F. da Silveira, F.B. Rizzato, Electromagnetic braking: a simple quantitative model, Am. J. Phys. 74 (9) (2006) 815–817, http://dx.doi.org/10.1119/1.2203645. [22] S. Cheah, H.A. Sodano, Novel eddy current damping mechanism for passive magnetic bearings, J. Vib. Control 14 (11) (2008) 1749–1766, http://dx.doi.org/10.1177/ 1077546308091219. [23] J. Bae, J. Hwang, J. Park, D. Kwag, Modeling and experiments on eddy current damping caused by a permanent magnet in a conductive tube, J. Mech. Sci. Technol. 23 (11) (2009) 3024–3035, http://dx.doi.org/10.1007/s12206-009-0819-0. [24] G. Donoso, C. Ladera, P. Martin, Magnet fall inside a conductive pipe: motion and the role of the pipe wall thickness, Eur. J. Phys. 30 (4) (2009) 855–869, http://dx. doi.org/10.1088/0143-0807/30/4/018. [25] G. Donoso, C. Ladera, P. Martin, Damped fall of magnets inside a conducting pipe, Am. J. Phys. 79 (2) (2011) 193–200, http://dx.doi.org/10.1119/1.3531964. [26] M. Symans, M. Constantinou, Passive fluid viscous damping systems for seismic energy dissipation, ISET J. Earthq. Technol. 35 (4) (1998) 185–206. [27] H.A. Sodano, J. Bae, D. Inman, W. Belvin, Concept and model of eddy current damper for vibration suppression of a beam, J. Sound Vib. 288 (4) (2005) 1177–1196, http://dx.doi.org/10.1016/j.jsv.2005.01.016. [28] B. Ebrahimi, M. Khamesee, M. Golnaraghi, Design and modeling of a magnetic shock absorber based on eddy current damping effect, J. Sound Vib. 315 (4) (2008) 875–889, http://dx.doi.org/10.1016/j.jsv.2008.02.022. [29] C. Elbuken, M. Khamesee, M. Yavuz, Eddy current damping for magnetic levitation: downscaling from macro-to micro-levitation, J. Phys. D: Appl. Phys. 39 (18) (2006) 3932–3938, http://dx.doi.org/10.1088/0022-3727/39/18/002. [30] J. Bae, J. Hwang, J. Roh, J. Kim, M. Yi, J. Lim, Vibration suppression of a cantilever beam using magnetically tuned-mass-damper, J. Sound Vib. 331 (26) (2012) 5669–5684, http://dx.doi.org/10.1016/j.jsv.2012.07.020. [31] E. Pestel, F. Leckie, Matrix Methods in Elastomechanics, 435, McGraw-Hill, New York, 1963. [32] J.T. Broch, Mechanical vibration and shock measurements, Brüel & Kjær, 1980. [33] L. Brillouin, Wave propagation in periodic structures: electric filters and crystal lattices, Cour. Corp. (2003). [34] Y. Levin, F.L. da Silveira, F.B. Rizzato, Electromagnetic braking: a simple quantitative model, Am. J. Phys. 74 (9) (2006) 815–817, http://dx.doi.org/10.1119/1.2203645. [35] R. Singh, V. Singh, Analysis of eddy current damper for suppression of vibrations using comsol software, Eng. Solid Mech. 3 (4) (2015) 215–222, http://dx.doi.org/ 10.5267/j.esm.2015.7.004. [36] M.A. Salam, Electromagnetic Field Theories for Engineering, Springer Science & Business Media; Singapore, 2014. [37] D.J. Inman, R.C. Singh, Engineering Vibration, 3, Prentice Hall; New Jersey, 2001.
Please cite this article as: E. Ozkaya, & C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.11.027i