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Shape optimization of phononic band gap structures incorporating inertial amplification mechanisms
Journal of Sound and Vibration 355 (2015) 232–245
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Shape optimization of phononic band gap structures incorporating inertial amplification mechanisms Osman Yuksel, Cetin Yilmaz n Department of Mechanical Engineering, Bogazici University, 34342 Bebek, Istanbul, Turkey
a r t i c l e in f o
abstract
Article history: Received 16 October 2014 Received in revised form 10 March 2015 Accepted 9 June 2015 Handling Editor: G. Degrande Available online 15 July 2015
The aim of this study is to obtain a wide and deep phononic gap at low frequencies in a two-dimensional periodic solid structure with embedded inertial amplification mechanisms. Size and shape optimizations are performed on the building block mechanism of the periodic structure to maximize phononic gap (stop band) width and depth. It is shown that shape optimization offers a wider and deeper gap, when both size and shape optimized mechanisms have the same mass and stiffness values. Analysis of the shape optimized mechanism is carried out using two different finite element models, one using beam elements and the other using shell elements. Both models produced similar results for the stop band width and depth. A two-dimensional periodic structure is constructed with the shape optimized building block mechanisms. Moreover, experimental and numerical frequency response results of this periodic structure are obtained. The matching frequency response results indicate that the two-dimensional periodic structure has a wide and deep phononic gap for in-plane excitations. Furthermore, due to proper selection of the out-of-plane thickness of the periodic structure, out-of-plane vibration modes do not occur within the phononic gap. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Inhibiting wave propagation in periodic structures, thus obtaining band gaps in which waves cannot propagate has long been studied [1]. When acoustic or elastic waves are considered, these gaps are called phononic band gaps. In the last decades, there are numerous studies that concern obtaining phononic band gaps in periodic media [2–10]. There exist two common methods to generate band gaps, namely, Bragg scattering and resonance scattering (local resonances). In Bragg scattering, band gap is created by using periodically varying mass and stiffness in the structures [11–13]. On the other hand, in resonance scattering, band gap is created by adding periodic local resonators to the structure [14–19]. Recently, inertial amplification method is proposed as a new band gap generation method [20,21]. In this method, band gap is engendered with amplification of a small mass by using displacement amplification mechanisms. In Ref. [21], finite periodic structures are studied and it is shown that inertial amplification induced gaps are qualitatively different from the two well-known gap types in terms of wave energy localization characteristics or gap depth profiles in frequency response function (FRF) plots. Moreover, Ref. [20] shows that by just increasing the amplification ratio, the band gap frequency range can be lowered without changing the total mass or stiffness of the structure. This is an advantage when the two common
n
Corresponding author. Tel.: þ90 212 359 6436. E-mail addresses: [email protected] (O. Yuksel), [email protected] (C. Yilmaz).
http://dx.doi.org/10.1016/j.jsv.2015.06.016 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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(y-x)cot(θ)/2
ma
y m
(x+y)/2 x
k
θ
m
Fig. 1. Lumped parameter model of the inertial amplification mechanism.
phononic band gap generation methods are considered. Since, to have a low frequency Bragg gap, heavy inclusions in a soft medium or a large lattice constant is needed. In addition, to obtain wide locally resonant gaps at low frequencies heavy resonators are required [20]. Phononic band gaps induced by inertial amplification are studied in two-dimensional lumped parameter systems [20,21], three dimensional lumped parameter lattices [22,23] and two-dimensional distributed parameter systems [24]. In the literature, shape and topology optimization have been used in phononic band gap structures to enhance gap properties [25–32]. However, shape optimization have not been used in two-dimensional inertially amplified phononic band gap structures, but only size optimization have been utilized [24]. In the current study, shape optimization will be used in two-dimensional distributed parameter systems to generate wider and deeper phononic gaps than that can be obtained by size optimization. By using a finite element model with shell elements, the numerical frequency response of the shape optimized phononic band gap structure with embedded inertial amplification mechanisms will be obtained. Moreover, this two-dimensional periodic structure will be manufactured and experimental validation of computational results will be done.
2. Analytical model The aim of this paper is to obtain wide and deep inertial amplification induced phononic gaps at low frequencies. For this purpose, a compliant mechanism will be designed via shape optimization as the building block of a two-dimensional periodic structure. Before analyzing the compliant inertial amplification building block mechanism, a simple lumped parameter mass–spring model, which is shown in Fig. 1, is examined. In this model, a spring with stiffness k is used to connect the two masses m, and inertial amplification mechanism is formed by the mass ma along with the two massless rigid links attached to it. As shown in Fig. 1, the motion of the mass ma can be acquired in terms of x, y and θ. In the case of small θ, the motion of mass ma would be amplified compared to the masses m. Consequently, this results in amplification of the effective inertia of the system [20,21]. The equation of motion of the lumped parameter inertial amplification mechanism, shown in Fig. 1, is given by [21] ðma ðcot2 θ þ 1Þ þ 4mÞx€ þ 4kx ¼ ma ðcot2 θ 1Þy€ þ4ky
(1)
In Eq. (1), when y is the input and x is the output, then the first resonance (ωp1) and antiresonance (ωz1) frequencies are found as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k (2) ωp1 ¼ m þ ma ðcot2 θ þ1Þ=4
ωz1 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ma ðcot2 θ 1Þ=4
(3)
In Eq. (2), static stiffness of the system is given by k, and dynamic mass, md, of the system is given by the denominator, which is md ¼ m þ
ma ðcot2 θ þ1Þ 4
(4)
Therefore, Eq. (2) can be simplified as
ωp1 ¼
sffiffiffiffiffiffiffi k md
(5)
Additionally, as can be seen in Fig. 1, the static mass, ms, of the system is given by ms ¼ 2m þ ma
(6)
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Thus, Eq. (3) can also be simplified as below noting that the denominator of Eq. (3) is equal to md ms =2: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ωz1 ¼ md ms =2
(7)
When examining Eqs. (5) and (7) it is seen that, as ms is greater than zero, ωp1 becomes less than ωz1. So, the system in Fig. 1 is a low pass filter type vibration isolator [33] and its displacement or acceleration transmissibility can be obtained by taking the Fourier transform of Eq. (1): TRðωÞ ¼
€ ωÞj j1 ðω=ωz1 Þ2 j jxðωÞj jxð ¼ ¼ € ωÞj j1 ðω=ωp1 Þ2 j jyðωÞj jyð
(8)
The frequency range where TRðωÞ o 1 is defined as the phononic gap (stop band). Moreover, the highest transmissibility value within the stop band is used to calculate the stop band depth. In Eq. (8), if the excitation frequency ω is greater than ωz1, then the transmissibility is less than ω2p1 =ω2z1 . As a result, band depth (BD) can be found as [24] BD ¼ 1 ω2p1 =ω2z1
(9)
Herein, according to Eq. (9), to increase band depth, ωp1 =ωz1 ratio should be decreased. The single degree-of-freedom system shown in Fig. 1 does not possess any second resonance frequency, which sets an upper limit for the stop band. On the other hand, a finite element model of a distributed parameter system will have multiple resonance frequencies. Therefore, in order to obtain a wide stop band, the interval between the first two resonance frequencies of the finite element model should be maximized. Moreover, the stop band limits of an inertial amplification mechanism and its one- or twodimensional arrays are expected to be the same [24]. Hence, as the stop band width of the building block mechanism is maximized, it is expected that the stop band widths of its one- or two-dimensional arrays will be maximized, as well. 3. Finite element models The distributed parameter model of the inertial amplification mechanism is shown in Fig. 2. Different sized rectangular beams are combined to form this mechanism. In Fig. 2, li and ti are, respectively, the length and the thickness of the ith beam which form the mechanism. Thicknesses t2 and t4 are much smaller than thicknesses t1 and t3. Thereby, the thin beams become compliant (flexure) hinges. In the first place, the finite element model of the distributed parameter system (compliant mechanism) shown in Fig. 2 is constructed by using one-dimensional offset beam elements. For this model, mass and stiffness matrices are multiplied by proper transformation matrices to take into consideration the eccentric connections of the beam elements that form the compliant mechanism [34]. Afterwards, a second finite element model is formed using shell elements in order to compare and verify the results. In the shell element model, a proper shell thickness is determined to ensure that the deformations remain in-plane. 4. Building block mechanism design The compliant mechanism in Fig. 2, which will be used as the building block of the phononic band gap structure is optimized to generate a wide and deep phononic gap at low frequencies. Initially, size optimization will be performed to specify the optimum sizes of the rectangular beams, which constitute the size optimized mechanism. After that, a separate shape optimization study will be conducted. In order to save computational time, the one-dimensional finite element model will be used during size and shape optimization processes. So as to attain a wide stop band at low frequencies, the interval between the first two resonance frequencies ðωp1 ; ωp2 Þ of the mechanism should be maximized. For that purpose, ωp2 =ωp1 ratio will be maximized. Additionally, as can be seen from Eq. (9), ωp1 =ωz1 ratio should be decreased to attain a deep stop band. By employing modal analysis, the first two resonance frequencies of the mechanism can be found quickly. Furthermore, by calculating the acceleration transmissibility of the mechanism, the first antiresonance frequency can be obtained. However, high frequency resolution is necessary to find the antiresonance frequency accurately, which leads to extra computational cost. As an alternative, Eq. (7) can be utilized to get a quite accurate approximation for the first antiresonance frequency. In the case for antiresonance, the displacement characteristics of the lumped and distributed parameter models
y
l1, t1
l2, t2
l3, t3
l3, t3
l4, t4
l1, t1
x
l2, t2
Fig. 2. Distributed parameter model of the inertial amplification mechanism.
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are similar. For a distributed parameter model, static stiffness of the system (k) can be calculated via finite element method. Moreover, the first resonance frequency of the system (ωp1) can be found by modal analysis. After that, by using Eq. (5), the dynamic mass of the system can be obtained as md ¼
k
ω2p1
(10)
The static mass of the system (ms) is the summation of the masses of the beam segments of the structure shown in Fig. 2. As a result, the first antiresonance frequency of the system can be obtained by utilizing Eq. (7). 4.1. Size optimization Size optimization is performed on the mechanism shown in Fig. 2. The objective function is chosen as maximization of the ratio between the second and the first resonance frequencies (max. ωp2 =ωp1 ). To obtain a stop band at low frequencies, the compliant hinge thicknesses (t2 and t4) should be as small as possible. The designed structure will be manufactured from a steel plate by wire—EDM (Electrical Discharge Machining) technique. t2 and t4 are taken as 0.5 mm, which is the smallest thickness that can be accurately produced. Moreover, increasing the length of the compliant hinges decreases the stiffness of the structure, which in turn shifts the stop band to lower frequencies. However, numerical studies show that increasing the length of the compliant hinges results in decreased stop band width. To have a considerably wide stop band at low frequencies l2 =t 2 ratio is selected as 5. In the first mode shape, the compliant hinges in the middle of the mechanism will bend twice in proportion to the hinges on the two ends. Hence, to obtain the same amount of stress in these hinges, l4 ¼ 2l2 is taken as a constraint. To form periodic structures, the designed mechanisms should be connected through the beam segments with length l1. Numerical studies show that decreasing the length of these beams results in increased stop band width. However, the length of these beams should not be very small, which would make the attachment of accelerometers or shakers difficult in testing. Consequently, the length l1 is chosen as 10 mm. The static stiffness of the compliant mechanism is constrained as k¼1000 kN/m. Depth of the stop band is set as a constraint as well, i.e., BD¼0.2. Moreover, elastic modulus of the material is E ¼205 GPa and density of the material is ρ ¼ 7800 kg/m3. Newton's method and genetic algorithm are used separately for size optimization problem. Both methods converged to the same results. The summary of the optimization problem formulation is given below: Maximize: ωp2 =ωp1 Subject to: kstatic ¼ 1000 kN=m BD ¼ 0:20 Fixed variables: t 2 ¼ t 4 ¼ 0:5 mm 2l2 ¼ l4 ¼ 5 mm l1 ¼ 10 mm Design variables: t 1 ; l3 ; t 3 As a result of optimization procedure, the design variables are found as follows: t 1 ¼ 4:1 mm, l3 ¼ 31:9 mm and t 3 ¼ 6:3 mm. 4.2. Shape optimization In order to improve stop band characteristics, as a next stage, shape optimization is performed. In the shape optimized design, all the beams in the mechanism shown in Fig. 2, except the beams with length l3, will be rectangular. When finding the optimal shape of these beams, t1, t3 and l3 will be recalculated. The same objective function, i.e., maximization of the ratio between the second and the first resonance frequencies (max. ωp2 =ωp1 ), is used for the shape optimization problem. In the size optimization study, when a larger band depth constraint was used, a narrower stop band was obtained. Therefore, it is challenging to obtain both wide and deep stop bands. For the shape optimization problem, a larger band depth constraint is chosen (BD ¼0.25). It is aimed to show that the shape optimized design can provide a wider stop band in the presence of this more challenging band depth constraint. Moreover, a mass constraint is also used for shape optimization problem to ensure that the shape and size optimized designs have the same mass value. In order to obtain the optimum shape of the beams with length l3, these beams are separated into 30 equal length segments. The shape optimized design will be used to form an equilateral triangle shaped structure, which will be the unit cell of a two-dimensional periodic structure (see Fig. 3). No further optimization will be performed as the two-dimensional periodic structure and its building block mechanism are expected to have stop bands in the same frequency range [24]. The dashed line in Fig. 3 shows the upper limit constraint for the thicknesses of the 30 segments. As a result, the mechanisms in the periodic structure will vibrate without contacting each other. Both Newton's method and genetic algorithm are utilized in the shape optimization problem. Again, these two different methods converged to the same results. The summary of the optimization problem formulation is given below: Maximize: ωp2 =ωp1 Subject to: kstatic ¼ 1000 kN=m
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l3 Fig. 3. Equilateral triangle unit cell block of the two-dimensional periodic structure. The dashed line shows the upper limit constraint for the thicknesses of the 30 segments of the beam with length l3.
BD ¼ 0:25 mðshape opt:Þ ¼ mðsize
opt:Þ
ðUpper limit constraint for the 30 thickness values of the segments that form the beam with length l3 as shown in Fig: 3Þ t 2 ¼ t 4 ¼ 0:5 mm 2l2 ¼ l4 ¼ 5 mm l1 ¼ 10 mm t 1 ; l3 and 30 thickness values of the segments that form the beam with length l3
Fixed variables:
Design variables:
As the result of the optimization procedure, the shape optimized design shown in Fig. 4(b) is obtained. The thickness t1 is found as 5.9 mm, the length l3 is found as 22.4 mm and the thicknesses of the 30 segments of the beam with length l3 can be seen in Fig. 4(b). 4.3. Numerical results The first three mode shapes of the size and shape optimized designs can be seen in Fig. 5. In general, the first three resonance frequencies of this mechanism can be written as ωp1 r ωp2 r ωp3 . As the gap between the first two resonance frequencies is maximized due to size optimization, ωp1 becomes small and ωp2 becomes large. However, ωp2 cannot be larger than ωp3. Therefore, as a result of maximization of the gap between the first two resonance frequencies, the second and the third resonance frequencies coincide. Yet, this is not the case for the shape optimization problem due to the extra mass constraint employed. The shape optimized design provides a wider and deeper stop band compared to size optimized design, considering that the total mass and stiffness are the same for both designs (see Table 1). So as to visualize the generated stop bands, acceleration transmissibilities of the size and shape optimized building block mechanisms are shown in Fig. 6. The horizontal input excitation with a small offset with respect to the centroidal axis of each mechanism is applied to observe the second and the third modes along with the first mode. As can be seen from Fig. 6, shape optimization yields a wider and deeper stop band considering that both designs have the same mass and stiffness values. To see the stop band characteristics in periodic structures, one-dimensional finite arrays of shape optimized mechanisms are formed by utilizing the one-dimensional finite element model. In Fig. 7, acceleration transmissibilities of finite arrays with two, four and six shape optimized mechanisms are shown. When this plot is investigated, it is seen that, band depth (see Eq. (9)) for two mechanisms is 0.61, for four mechanisms it is more than 0.91 and for six mechanisms it is more than 0.98. That is to say; if four shape optimized mechanisms are used in the array, less than 9 percent, if six shape optimized mechanisms are used in the array, less than 2 percent of the input vibration is transmitted to the output end. In all the transmissibility plots, the first stop band starts at the same frequency where TR ¼1 line is crossed. The lower limit of the stop band is determined by this frequency, which is denoted as ωs. The first resonance and antiresonance frequencies of the shape optimized mechanism are much smaller than the other resonance and antiresonance frequencies. Therefore, the acceleration transmissibility of shape optimized compliant mechanism can be very closely approximated by
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20
20
0
0
−20
−20 −40
−20
0
20
40
−40
−20
0
237
20
40
Fig. 4. (a) Size optimized building block mechanism (ωp2 =ωp1 ¼ 2:79, BD ¼0.20). (b) Shape optimized building block mechanism (ωp2 =ωp1 ¼ 3:32, BD ¼0.25).
Mode Shape 1 (250.0 Hz)
Mode Shape 1 (283.2 Hz)
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0
−20
−20 −40
−20
0
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40
−40
Mode Shape 2 (697.8 Hz)
−20
0
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40
Mode Shape 2 (941.1 Hz)
20
20
0
0
−20
−20 −40
−20
0
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−40
Mode Shape 3 (697.8 Hz)
−20
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Mode Shape 3 (1069.1 Hz)
20
20
0
0
−20
−20 −40
−20
0
20
40
−40
−20
0
20
40
Fig. 5. First three mode shapes of the (a) size optimized building block mechanism and (b) shape optimized building block mechanism. Table 1 Comparison of size and shape optimized building block mechanism designs.
ωp2/ωp1 ratio Band depth (BD) Mass (kg) Stiffness (kN/m)
Size optimized design
Shape optimized design
2.79 0.20 0.143 1000
3.32 0.25 0.143 1000
the lumped parameter system's transmissibility formulation shown in Eq. (8). Hence, by equating Eq. (8) to 1, solved [24]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2ω2 ω2 ωs ¼ t 2 p1 z12 ωp1 þ ωz1
ωs can be
(11)
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Transmissibility
10
0
10
−1
10
0
200
400
600
800
1000
1200
Frequency (Hz) Fig. 6. Acceleration transmissibility plots of size and shape optimized building block mechanism designs.
2
10
s s s
1
Transmissibility
10
0
10
−1
10
−2
10
−3
10
0
200
400
600
800
1000
1200
Frequency (Hz) Fig. 7. Acceleration transmissibility plots of one-dimensional finite arrays of shape optimized building block mechanisms.
For the shape optimized compliant mechanism, the first resonance frequency (ωp1) occurs at 283.2 Hz, and the first antiresonance frequency (ωz1) occurs at 322.6 Hz. Then ωs is calculated as 301 Hz by using Eq. (11). The ωs value obtained from acceleration transmissibility plots is 300.9 Hz which is almost the same value obtained using Eq. (11). Furthermore, the stop band ends at the second resonance frequency of the structure (ωp2) which occurs at 941.1 Hz. Phononic band structure of the infinite periodic shape optimized compliant mechanism is shown in Fig. 8. As can be seen in Fig. 8, the first band gap is formed between 300.9 Hz and 941.1 Hz. Consequently, stop limits for the finite and infinite one-dimensional periodic cases are in agreement with each other. In order to check the proposed one-dimensional finite element model's validity, a second model that is shown in Fig. 9 is formed by using shell elements. In this finite element model shape of the beams with length l3 is smoothened for the ease of manufacturing. Poisson ratio is taken as 0.29. Moreover, shell thickness is taken as 55 mm to prevent any out-of-plane mode to occur within stop band of the periodic structure. Acceleration transmissibilities of shape optimized beam and shell element models are compared in Fig. 11. One can see that the difference between the first resonance frequencies is 10.8 percent, the second resonance frequencies is 15.1 percent, the first antiresonance frequencies is 11.1 percent. These differences stem from element type change and shape smoothening for manufacturing purposes. Nevertheless, stop band width and depth depend on ωp2 =ωp1 and ωp1 =ωz1 ratios, respectively. The difference for the ωp2 =ωp1 ratio is 3.8 percent and the difference for the ωp1 =ωz1 ratio is 0.3 percent. Thus, if shell elements were used in the shape optimization problem, similar results would be obtained but this would be computationally expensive.
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Band Structure 1200
1000
Frequency (Hz)
800
600
400
200
0
0
0.2
0.4
0.6
0.8
1
Wave vector (xπ/l t ) Fig. 8. Band structure of the shape optimized building block mechanism. Here, lt represents length of the mechanism.
Fig. 9. Shell element model of the shape optimized building block mechanism.
Fig. 10. First two modes of the shell element model of the shape optimized building block mechanism. (a) The first mode (255.7 Hz) and (b) the second mode (817.4 Hz).
4.4. Pareto study In this section, the relation between the stop band width and depth is investigated via two sets of Pareto curves. In order to have a fair comparison among the curves, mass and stiffness values are taken as 0.143 kg and 1000 kN/m, respectively. Since, the first stop band is formed between ωs and ωp2, then normalized band width (BW) can be introduced as in the following equation:
ωp2 ωs BW ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωp2 ωs
(12)
This band width definition can be used as the objective function in the optimization problem to obtain wide and deep gaps. Therefore, in the size and shape optimization problems, instead of maximizing ωp2 =ωp1 one can maximize BW. First set of Pareto curves given in Fig. 12 shows the relationship between band width (BW) (Eq. (12)) and band depth (BD)
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Transmissibility
10
Shell element model Beam element model
0
10
−1
10
0
200
400
600 Frequency (Hz)
800
1000
1200
Fig. 11. Comparison of acceleration transmissibilities of one-dimensional offset beam element and shell element shape optimized building block mechanisms.
1.7 1.6 1.5
Band width (BW)
1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.1
0.15
0.2
0.25
0.3
0.35
0.4
Band depth (BD) Fig. 12. Band width versus band depth of optimized mechanism designs. Red cross: selected size optimized design (BD ¼ 0.20, BW ¼1.02) and blue circle: selected shape optimized design (BD ¼0.25, BW ¼1.20). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
(Eq. (9)). As can be seen from Fig. 12, there is an inverse relationship between band width and band depth. It can also be noted that the Pareto curve of the size optimized designs lies well below the Pareto curve of the shape optimized designs. In the second set of Pareto curves, the relationship between ωp2 =ωp1 ratio and band depth (BD) is investigated (see Fig. 13). Again, an inverse relationship exists, that is to say, in order to have wider stop bands, band depth is sacrificed and vice versa. In this figure, the Pareto curve of the size optimized designs again lies well below the Pareto curve of the shape optimized designs. In other words, utilizing shape optimization instead of size optimization provides wider or deeper stop bands for the same band depth or width value. 5. Two-dimensional phononic band gap structure: experimental and numerical results The shape optimized mechanism which is depicted in Fig. 9 is used as the building block for the phononic band gap structure (see Fig. 14). By using shell elements, natural frequencies and mode shapes of this periodic structure are found for free boundary conditions. 29 modes are calculated up to 264.6 Hz. Yet, the next mode appears at 829.5 Hz. Therefore, the first stop band of this periodic structure is generated between these two frequencies. The two modes that set the limit for the stop band can be seen in Figs. 15 and 16. Notice that, in Fig. 15 the building block mechanisms deform similar to the first mode shape given in Fig. 10(a). Moreover, in Fig. 16 these mechanisms deform similar to their second mode shape as shown
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4.5
3.5
ωp2/ω
p1
4
3
2.5
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Band depth (BD) Fig. 13. ωp2 =ωp1 ratio versus band depth of optimized mechanism designs. Red cross: selected size optimized design (BD ¼ 0.20, ωp2 =ωp1 ¼2.79) and blue circle: selected shape optimized design (BD ¼ 0.25, ωp2 =ωp1 ¼ 3.32). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 14. Two-dimensional phononic band gap structure.
Fig. 15. 29th mode shape of the phononic band gap structure (264.5 Hz).
Fig. 16. 30th mode shape of the phononic band gap structure (829.5 Hz).
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Fig. 17. Manufactured phononic band gap structure.
Fig. 18. Experimental setup.
3
10
2
(ω)/y(ω)|
10
Transmissibility,
:
1
10
1
:
0
10
−1
10
−2
10
−3
10
−4
10
100
200
300
400
500
600
700
800
900
1000
Frequency (Hz) Fig. 19. Experimental and numerical acceleration transmissibility plots of the phononic band gap structure for the longitudinal direction (x1).
in Fig. 10(b). Therefore, the first two modes of the building block mechanism define the limits of the first stop band for the two-dimensional periodic structure. The out-of-plane thickness of the structure does not affect the frequency response for in-plane excitations. Therefore, the stop band frequency range is not a function of out-of-plane thickness of the structure. However, the out-of-plane bending modes of the structure are affected by this thickness. Numerical modal analysis results show that 55 mm is the minimum thickness that places the first out-of-plane bending mode above the stop band frequency range. In order to have a compact specimen that can be accurately manufactured, the out-of-plane thickness is selected as 55 mm. The periodic structure shown in Fig. 14 is manufactured via wire-EDM method (see Fig. 17). The material properties of the sample are the same with the numerical model (E ¼205 GPa, ρ ¼ 7800 kg/m3, ν ¼ 0:29).
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10
..
243
..
|x (ω)/y(ω)| 1
..
..
|x2(ω)/y(ω)|
10
Transmissibility
10
10
10
10 x (ω)
10
x (ω) y(ω)
10
100
200
300
400 500 600 Frequency (Hz)
700
800
900
1000
Fig. 20. Experimental acceleration transmissibility plots of the phononic band gap structure for longitudinal (x1) and transverse (x2) directions.
2
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.. .. .. .. |x (ω)/y(ω)| |x1(ω)/y(ω)|
1
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0
Transmissibility
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1000
Fig. 21. Experimental acceleration transmissibility plots of the phononic band gap structure for longitudinal (x1) and out-of-plane (x3) directions.
In the experimental study, the shape optimized periodic steel structure that is supported by rubber cords is excited from one end via a modal shaker (see Fig. 18). The input acceleration is measured by an impedance head and the output acceleration is measured by an accelerometer. As the noise level of the impedance head is smaller than that of the output accelerometer, H1 estimate [35] of the acceleration transmissibility frequency response function is used. The average of 90 measurements is calculated to reduce the effect of noise in the system. Moreover, Hanning window is utilized to reduce spectral leakage. In Fig. 19, finite element analysis and experimental frequency response results in longitudinal direction are compared. Here, the input acceleration is y€ and the output acceleration is x€1 . It can be seen that experimental and finite elements analysis results are consistent with each other and the stop band forms in the same frequency range (265–830 Hz). This graph also shows that, within the stop band, less than 0.5 percent of input vibration is transmitted to the output end. In Fig. 20, experimental results for transmissibilities for longitudinal and transverse directions are compared. For this case, the in-plane input makes 451 with respect to the x1- and x2-axes, while accelerations in longitudinal (x1) and transverse (x2) directions are observed as outputs. Less than 1 percent of input vibration is transmitted to the output end for this case. Again stop band limits are the same as before (265–830 Hz). Therefore, Figs. 19 and 20 indicate that the twodimensional periodic structure, which is formed by using the shape optimized building block mechanisms, provides a wide (between 265 and 830 Hz) and deep (BD 4 0:99) stop band. The experimental transmissibility results for the out-of-plane direction are given in Fig. 21. For this case, a longitudinal input with a slight out-of-plane inclination is fed to the structure while the accelerations in the out-of-plane (x3) and
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longitudinal (x1) directions are observed as outputs. It can be seen that by choosing 55 mm thickness for the steel sample, out-of-plane modes of vibration are avoided in the stop band frequency range. 6. Conclusion In this paper, a two-dimensional periodic structure, which has a wide and deep stop band at low frequencies is designed via inertial amplification method. Size and shape optimizations are performed on the one-dimensional beam element model of the building block mechanism to achieve wide and deep stop bands. For the same stiffness and mass values, the shape optimized design provides a wider and deeper stop band compared to the size optimized one. Besides, the relationship between band width and band depth is shown in Pareto studies in which shape optimization procedure provides better designs than that can be obtained by size optimization. Moreover, a shell element model of the building block mechanism is formed and this model shows similar stop band characteristics as the beam element model, indicating that the onedimensional finite element model is valid for a fast optimization procedure. Using the shell element model, a twodimensional periodic structure is formed. It is shown that the stop band limits of this periodic structure are nearly the same with its building block mechanism. Moreover, the two-dimensional phononic band gap structure is manufactured with a suitable out-of-plane thickness to prevent any out-of-plane vibration modes to occur within the stop band. Frequency response of the phononic band gap structure is obtained numerically and experimentally and it is found that they are close to each other. Experimental results show that, a wide and deep phononic gap is created for in-plane excitations in both longitudinal and transverse directions. It is also shown via experiments that out-of-plane vibration modes do not occur within the phononic gap due to the proper selection of the out-of-plane thickness of the periodic structure.
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Shape optimization of phononic band gap structures incorporating inertial amplification mechanisms Osman Yuksel, Cetin Yilmaz n Department of Mechanical Engineering, Bogazici University, 34342 Bebek, Istanbul, Turkey
a r t i c l e in f o
abstract
Article history: Received 16 October 2014 Received in revised form 10 March 2015 Accepted 9 June 2015 Handling Editor: G. Degrande Available online 15 July 2015
The aim of this study is to obtain a wide and deep phononic gap at low frequencies in a two-dimensional periodic solid structure with embedded inertial amplification mechanisms. Size and shape optimizations are performed on the building block mechanism of the periodic structure to maximize phononic gap (stop band) width and depth. It is shown that shape optimization offers a wider and deeper gap, when both size and shape optimized mechanisms have the same mass and stiffness values. Analysis of the shape optimized mechanism is carried out using two different finite element models, one using beam elements and the other using shell elements. Both models produced similar results for the stop band width and depth. A two-dimensional periodic structure is constructed with the shape optimized building block mechanisms. Moreover, experimental and numerical frequency response results of this periodic structure are obtained. The matching frequency response results indicate that the two-dimensional periodic structure has a wide and deep phononic gap for in-plane excitations. Furthermore, due to proper selection of the out-of-plane thickness of the periodic structure, out-of-plane vibration modes do not occur within the phononic gap. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Inhibiting wave propagation in periodic structures, thus obtaining band gaps in which waves cannot propagate has long been studied [1]. When acoustic or elastic waves are considered, these gaps are called phononic band gaps. In the last decades, there are numerous studies that concern obtaining phononic band gaps in periodic media [2–10]. There exist two common methods to generate band gaps, namely, Bragg scattering and resonance scattering (local resonances). In Bragg scattering, band gap is created by using periodically varying mass and stiffness in the structures [11–13]. On the other hand, in resonance scattering, band gap is created by adding periodic local resonators to the structure [14–19]. Recently, inertial amplification method is proposed as a new band gap generation method [20,21]. In this method, band gap is engendered with amplification of a small mass by using displacement amplification mechanisms. In Ref. [21], finite periodic structures are studied and it is shown that inertial amplification induced gaps are qualitatively different from the two well-known gap types in terms of wave energy localization characteristics or gap depth profiles in frequency response function (FRF) plots. Moreover, Ref. [20] shows that by just increasing the amplification ratio, the band gap frequency range can be lowered without changing the total mass or stiffness of the structure. This is an advantage when the two common
n
Corresponding author. Tel.: þ90 212 359 6436. E-mail addresses: [email protected] (O. Yuksel), [email protected] (C. Yilmaz).
http://dx.doi.org/10.1016/j.jsv.2015.06.016 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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233
(y-x)cot(θ)/2
ma
y m
(x+y)/2 x
k
θ
m
Fig. 1. Lumped parameter model of the inertial amplification mechanism.
phononic band gap generation methods are considered. Since, to have a low frequency Bragg gap, heavy inclusions in a soft medium or a large lattice constant is needed. In addition, to obtain wide locally resonant gaps at low frequencies heavy resonators are required [20]. Phononic band gaps induced by inertial amplification are studied in two-dimensional lumped parameter systems [20,21], three dimensional lumped parameter lattices [22,23] and two-dimensional distributed parameter systems [24]. In the literature, shape and topology optimization have been used in phononic band gap structures to enhance gap properties [25–32]. However, shape optimization have not been used in two-dimensional inertially amplified phononic band gap structures, but only size optimization have been utilized [24]. In the current study, shape optimization will be used in two-dimensional distributed parameter systems to generate wider and deeper phononic gaps than that can be obtained by size optimization. By using a finite element model with shell elements, the numerical frequency response of the shape optimized phononic band gap structure with embedded inertial amplification mechanisms will be obtained. Moreover, this two-dimensional periodic structure will be manufactured and experimental validation of computational results will be done.
2. Analytical model The aim of this paper is to obtain wide and deep inertial amplification induced phononic gaps at low frequencies. For this purpose, a compliant mechanism will be designed via shape optimization as the building block of a two-dimensional periodic structure. Before analyzing the compliant inertial amplification building block mechanism, a simple lumped parameter mass–spring model, which is shown in Fig. 1, is examined. In this model, a spring with stiffness k is used to connect the two masses m, and inertial amplification mechanism is formed by the mass ma along with the two massless rigid links attached to it. As shown in Fig. 1, the motion of the mass ma can be acquired in terms of x, y and θ. In the case of small θ, the motion of mass ma would be amplified compared to the masses m. Consequently, this results in amplification of the effective inertia of the system [20,21]. The equation of motion of the lumped parameter inertial amplification mechanism, shown in Fig. 1, is given by [21] ðma ðcot2 θ þ 1Þ þ 4mÞx€ þ 4kx ¼ ma ðcot2 θ 1Þy€ þ4ky
(1)
In Eq. (1), when y is the input and x is the output, then the first resonance (ωp1) and antiresonance (ωz1) frequencies are found as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k (2) ωp1 ¼ m þ ma ðcot2 θ þ1Þ=4
ωz1 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ma ðcot2 θ 1Þ=4
(3)
In Eq. (2), static stiffness of the system is given by k, and dynamic mass, md, of the system is given by the denominator, which is md ¼ m þ
ma ðcot2 θ þ1Þ 4
(4)
Therefore, Eq. (2) can be simplified as
ωp1 ¼
sffiffiffiffiffiffiffi k md
(5)
Additionally, as can be seen in Fig. 1, the static mass, ms, of the system is given by ms ¼ 2m þ ma
(6)
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Thus, Eq. (3) can also be simplified as below noting that the denominator of Eq. (3) is equal to md ms =2: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ωz1 ¼ md ms =2
(7)
When examining Eqs. (5) and (7) it is seen that, as ms is greater than zero, ωp1 becomes less than ωz1. So, the system in Fig. 1 is a low pass filter type vibration isolator [33] and its displacement or acceleration transmissibility can be obtained by taking the Fourier transform of Eq. (1): TRðωÞ ¼
€ ωÞj j1 ðω=ωz1 Þ2 j jxðωÞj jxð ¼ ¼ € ωÞj j1 ðω=ωp1 Þ2 j jyðωÞj jyð
(8)
The frequency range where TRðωÞ o 1 is defined as the phononic gap (stop band). Moreover, the highest transmissibility value within the stop band is used to calculate the stop band depth. In Eq. (8), if the excitation frequency ω is greater than ωz1, then the transmissibility is less than ω2p1 =ω2z1 . As a result, band depth (BD) can be found as [24] BD ¼ 1 ω2p1 =ω2z1
(9)
Herein, according to Eq. (9), to increase band depth, ωp1 =ωz1 ratio should be decreased. The single degree-of-freedom system shown in Fig. 1 does not possess any second resonance frequency, which sets an upper limit for the stop band. On the other hand, a finite element model of a distributed parameter system will have multiple resonance frequencies. Therefore, in order to obtain a wide stop band, the interval between the first two resonance frequencies of the finite element model should be maximized. Moreover, the stop band limits of an inertial amplification mechanism and its one- or twodimensional arrays are expected to be the same [24]. Hence, as the stop band width of the building block mechanism is maximized, it is expected that the stop band widths of its one- or two-dimensional arrays will be maximized, as well. 3. Finite element models The distributed parameter model of the inertial amplification mechanism is shown in Fig. 2. Different sized rectangular beams are combined to form this mechanism. In Fig. 2, li and ti are, respectively, the length and the thickness of the ith beam which form the mechanism. Thicknesses t2 and t4 are much smaller than thicknesses t1 and t3. Thereby, the thin beams become compliant (flexure) hinges. In the first place, the finite element model of the distributed parameter system (compliant mechanism) shown in Fig. 2 is constructed by using one-dimensional offset beam elements. For this model, mass and stiffness matrices are multiplied by proper transformation matrices to take into consideration the eccentric connections of the beam elements that form the compliant mechanism [34]. Afterwards, a second finite element model is formed using shell elements in order to compare and verify the results. In the shell element model, a proper shell thickness is determined to ensure that the deformations remain in-plane. 4. Building block mechanism design The compliant mechanism in Fig. 2, which will be used as the building block of the phononic band gap structure is optimized to generate a wide and deep phononic gap at low frequencies. Initially, size optimization will be performed to specify the optimum sizes of the rectangular beams, which constitute the size optimized mechanism. After that, a separate shape optimization study will be conducted. In order to save computational time, the one-dimensional finite element model will be used during size and shape optimization processes. So as to attain a wide stop band at low frequencies, the interval between the first two resonance frequencies ðωp1 ; ωp2 Þ of the mechanism should be maximized. For that purpose, ωp2 =ωp1 ratio will be maximized. Additionally, as can be seen from Eq. (9), ωp1 =ωz1 ratio should be decreased to attain a deep stop band. By employing modal analysis, the first two resonance frequencies of the mechanism can be found quickly. Furthermore, by calculating the acceleration transmissibility of the mechanism, the first antiresonance frequency can be obtained. However, high frequency resolution is necessary to find the antiresonance frequency accurately, which leads to extra computational cost. As an alternative, Eq. (7) can be utilized to get a quite accurate approximation for the first antiresonance frequency. In the case for antiresonance, the displacement characteristics of the lumped and distributed parameter models
y
l1, t1
l2, t2
l3, t3
l3, t3
l4, t4
l1, t1
x
l2, t2
Fig. 2. Distributed parameter model of the inertial amplification mechanism.
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are similar. For a distributed parameter model, static stiffness of the system (k) can be calculated via finite element method. Moreover, the first resonance frequency of the system (ωp1) can be found by modal analysis. After that, by using Eq. (5), the dynamic mass of the system can be obtained as md ¼
k
ω2p1
(10)
The static mass of the system (ms) is the summation of the masses of the beam segments of the structure shown in Fig. 2. As a result, the first antiresonance frequency of the system can be obtained by utilizing Eq. (7). 4.1. Size optimization Size optimization is performed on the mechanism shown in Fig. 2. The objective function is chosen as maximization of the ratio between the second and the first resonance frequencies (max. ωp2 =ωp1 ). To obtain a stop band at low frequencies, the compliant hinge thicknesses (t2 and t4) should be as small as possible. The designed structure will be manufactured from a steel plate by wire—EDM (Electrical Discharge Machining) technique. t2 and t4 are taken as 0.5 mm, which is the smallest thickness that can be accurately produced. Moreover, increasing the length of the compliant hinges decreases the stiffness of the structure, which in turn shifts the stop band to lower frequencies. However, numerical studies show that increasing the length of the compliant hinges results in decreased stop band width. To have a considerably wide stop band at low frequencies l2 =t 2 ratio is selected as 5. In the first mode shape, the compliant hinges in the middle of the mechanism will bend twice in proportion to the hinges on the two ends. Hence, to obtain the same amount of stress in these hinges, l4 ¼ 2l2 is taken as a constraint. To form periodic structures, the designed mechanisms should be connected through the beam segments with length l1. Numerical studies show that decreasing the length of these beams results in increased stop band width. However, the length of these beams should not be very small, which would make the attachment of accelerometers or shakers difficult in testing. Consequently, the length l1 is chosen as 10 mm. The static stiffness of the compliant mechanism is constrained as k¼1000 kN/m. Depth of the stop band is set as a constraint as well, i.e., BD¼0.2. Moreover, elastic modulus of the material is E ¼205 GPa and density of the material is ρ ¼ 7800 kg/m3. Newton's method and genetic algorithm are used separately for size optimization problem. Both methods converged to the same results. The summary of the optimization problem formulation is given below: Maximize: ωp2 =ωp1 Subject to: kstatic ¼ 1000 kN=m BD ¼ 0:20 Fixed variables: t 2 ¼ t 4 ¼ 0:5 mm 2l2 ¼ l4 ¼ 5 mm l1 ¼ 10 mm Design variables: t 1 ; l3 ; t 3 As a result of optimization procedure, the design variables are found as follows: t 1 ¼ 4:1 mm, l3 ¼ 31:9 mm and t 3 ¼ 6:3 mm. 4.2. Shape optimization In order to improve stop band characteristics, as a next stage, shape optimization is performed. In the shape optimized design, all the beams in the mechanism shown in Fig. 2, except the beams with length l3, will be rectangular. When finding the optimal shape of these beams, t1, t3 and l3 will be recalculated. The same objective function, i.e., maximization of the ratio between the second and the first resonance frequencies (max. ωp2 =ωp1 ), is used for the shape optimization problem. In the size optimization study, when a larger band depth constraint was used, a narrower stop band was obtained. Therefore, it is challenging to obtain both wide and deep stop bands. For the shape optimization problem, a larger band depth constraint is chosen (BD ¼0.25). It is aimed to show that the shape optimized design can provide a wider stop band in the presence of this more challenging band depth constraint. Moreover, a mass constraint is also used for shape optimization problem to ensure that the shape and size optimized designs have the same mass value. In order to obtain the optimum shape of the beams with length l3, these beams are separated into 30 equal length segments. The shape optimized design will be used to form an equilateral triangle shaped structure, which will be the unit cell of a two-dimensional periodic structure (see Fig. 3). No further optimization will be performed as the two-dimensional periodic structure and its building block mechanism are expected to have stop bands in the same frequency range [24]. The dashed line in Fig. 3 shows the upper limit constraint for the thicknesses of the 30 segments. As a result, the mechanisms in the periodic structure will vibrate without contacting each other. Both Newton's method and genetic algorithm are utilized in the shape optimization problem. Again, these two different methods converged to the same results. The summary of the optimization problem formulation is given below: Maximize: ωp2 =ωp1 Subject to: kstatic ¼ 1000 kN=m
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30
l3 Fig. 3. Equilateral triangle unit cell block of the two-dimensional periodic structure. The dashed line shows the upper limit constraint for the thicknesses of the 30 segments of the beam with length l3.
BD ¼ 0:25 mðshape opt:Þ ¼ mðsize
opt:Þ
ðUpper limit constraint for the 30 thickness values of the segments that form the beam with length l3 as shown in Fig: 3Þ t 2 ¼ t 4 ¼ 0:5 mm 2l2 ¼ l4 ¼ 5 mm l1 ¼ 10 mm t 1 ; l3 and 30 thickness values of the segments that form the beam with length l3
Fixed variables:
Design variables:
As the result of the optimization procedure, the shape optimized design shown in Fig. 4(b) is obtained. The thickness t1 is found as 5.9 mm, the length l3 is found as 22.4 mm and the thicknesses of the 30 segments of the beam with length l3 can be seen in Fig. 4(b). 4.3. Numerical results The first three mode shapes of the size and shape optimized designs can be seen in Fig. 5. In general, the first three resonance frequencies of this mechanism can be written as ωp1 r ωp2 r ωp3 . As the gap between the first two resonance frequencies is maximized due to size optimization, ωp1 becomes small and ωp2 becomes large. However, ωp2 cannot be larger than ωp3. Therefore, as a result of maximization of the gap between the first two resonance frequencies, the second and the third resonance frequencies coincide. Yet, this is not the case for the shape optimization problem due to the extra mass constraint employed. The shape optimized design provides a wider and deeper stop band compared to size optimized design, considering that the total mass and stiffness are the same for both designs (see Table 1). So as to visualize the generated stop bands, acceleration transmissibilities of the size and shape optimized building block mechanisms are shown in Fig. 6. The horizontal input excitation with a small offset with respect to the centroidal axis of each mechanism is applied to observe the second and the third modes along with the first mode. As can be seen from Fig. 6, shape optimization yields a wider and deeper stop band considering that both designs have the same mass and stiffness values. To see the stop band characteristics in periodic structures, one-dimensional finite arrays of shape optimized mechanisms are formed by utilizing the one-dimensional finite element model. In Fig. 7, acceleration transmissibilities of finite arrays with two, four and six shape optimized mechanisms are shown. When this plot is investigated, it is seen that, band depth (see Eq. (9)) for two mechanisms is 0.61, for four mechanisms it is more than 0.91 and for six mechanisms it is more than 0.98. That is to say; if four shape optimized mechanisms are used in the array, less than 9 percent, if six shape optimized mechanisms are used in the array, less than 2 percent of the input vibration is transmitted to the output end. In all the transmissibility plots, the first stop band starts at the same frequency where TR ¼1 line is crossed. The lower limit of the stop band is determined by this frequency, which is denoted as ωs. The first resonance and antiresonance frequencies of the shape optimized mechanism are much smaller than the other resonance and antiresonance frequencies. Therefore, the acceleration transmissibility of shape optimized compliant mechanism can be very closely approximated by
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20
20
0
0
−20
−20 −40
−20
0
20
40
−40
−20
0
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20
40
Fig. 4. (a) Size optimized building block mechanism (ωp2 =ωp1 ¼ 2:79, BD ¼0.20). (b) Shape optimized building block mechanism (ωp2 =ωp1 ¼ 3:32, BD ¼0.25).
Mode Shape 1 (250.0 Hz)
Mode Shape 1 (283.2 Hz)
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Fig. 5. First three mode shapes of the (a) size optimized building block mechanism and (b) shape optimized building block mechanism. Table 1 Comparison of size and shape optimized building block mechanism designs.
ωp2/ωp1 ratio Band depth (BD) Mass (kg) Stiffness (kN/m)
Size optimized design
Shape optimized design
2.79 0.20 0.143 1000
3.32 0.25 0.143 1000
the lumped parameter system's transmissibility formulation shown in Eq. (8). Hence, by equating Eq. (8) to 1, solved [24]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2ω2 ω2 ωs ¼ t 2 p1 z12 ωp1 þ ωz1
ωs can be
(11)
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Transmissibility
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Frequency (Hz) Fig. 6. Acceleration transmissibility plots of size and shape optimized building block mechanism designs.
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Frequency (Hz) Fig. 7. Acceleration transmissibility plots of one-dimensional finite arrays of shape optimized building block mechanisms.
For the shape optimized compliant mechanism, the first resonance frequency (ωp1) occurs at 283.2 Hz, and the first antiresonance frequency (ωz1) occurs at 322.6 Hz. Then ωs is calculated as 301 Hz by using Eq. (11). The ωs value obtained from acceleration transmissibility plots is 300.9 Hz which is almost the same value obtained using Eq. (11). Furthermore, the stop band ends at the second resonance frequency of the structure (ωp2) which occurs at 941.1 Hz. Phononic band structure of the infinite periodic shape optimized compliant mechanism is shown in Fig. 8. As can be seen in Fig. 8, the first band gap is formed between 300.9 Hz and 941.1 Hz. Consequently, stop limits for the finite and infinite one-dimensional periodic cases are in agreement with each other. In order to check the proposed one-dimensional finite element model's validity, a second model that is shown in Fig. 9 is formed by using shell elements. In this finite element model shape of the beams with length l3 is smoothened for the ease of manufacturing. Poisson ratio is taken as 0.29. Moreover, shell thickness is taken as 55 mm to prevent any out-of-plane mode to occur within stop band of the periodic structure. Acceleration transmissibilities of shape optimized beam and shell element models are compared in Fig. 11. One can see that the difference between the first resonance frequencies is 10.8 percent, the second resonance frequencies is 15.1 percent, the first antiresonance frequencies is 11.1 percent. These differences stem from element type change and shape smoothening for manufacturing purposes. Nevertheless, stop band width and depth depend on ωp2 =ωp1 and ωp1 =ωz1 ratios, respectively. The difference for the ωp2 =ωp1 ratio is 3.8 percent and the difference for the ωp1 =ωz1 ratio is 0.3 percent. Thus, if shell elements were used in the shape optimization problem, similar results would be obtained but this would be computationally expensive.
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Band Structure 1200
1000
Frequency (Hz)
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Wave vector (xπ/l t ) Fig. 8. Band structure of the shape optimized building block mechanism. Here, lt represents length of the mechanism.
Fig. 9. Shell element model of the shape optimized building block mechanism.
Fig. 10. First two modes of the shell element model of the shape optimized building block mechanism. (a) The first mode (255.7 Hz) and (b) the second mode (817.4 Hz).
4.4. Pareto study In this section, the relation between the stop band width and depth is investigated via two sets of Pareto curves. In order to have a fair comparison among the curves, mass and stiffness values are taken as 0.143 kg and 1000 kN/m, respectively. Since, the first stop band is formed between ωs and ωp2, then normalized band width (BW) can be introduced as in the following equation:
ωp2 ωs BW ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωp2 ωs
(12)
This band width definition can be used as the objective function in the optimization problem to obtain wide and deep gaps. Therefore, in the size and shape optimization problems, instead of maximizing ωp2 =ωp1 one can maximize BW. First set of Pareto curves given in Fig. 12 shows the relationship between band width (BW) (Eq. (12)) and band depth (BD)
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Shell element model Beam element model
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800
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Fig. 11. Comparison of acceleration transmissibilities of one-dimensional offset beam element and shell element shape optimized building block mechanisms.
1.7 1.6 1.5
Band width (BW)
1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.1
0.15
0.2
0.25
0.3
0.35
0.4
Band depth (BD) Fig. 12. Band width versus band depth of optimized mechanism designs. Red cross: selected size optimized design (BD ¼ 0.20, BW ¼1.02) and blue circle: selected shape optimized design (BD ¼0.25, BW ¼1.20). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
(Eq. (9)). As can be seen from Fig. 12, there is an inverse relationship between band width and band depth. It can also be noted that the Pareto curve of the size optimized designs lies well below the Pareto curve of the shape optimized designs. In the second set of Pareto curves, the relationship between ωp2 =ωp1 ratio and band depth (BD) is investigated (see Fig. 13). Again, an inverse relationship exists, that is to say, in order to have wider stop bands, band depth is sacrificed and vice versa. In this figure, the Pareto curve of the size optimized designs again lies well below the Pareto curve of the shape optimized designs. In other words, utilizing shape optimization instead of size optimization provides wider or deeper stop bands for the same band depth or width value. 5. Two-dimensional phononic band gap structure: experimental and numerical results The shape optimized mechanism which is depicted in Fig. 9 is used as the building block for the phononic band gap structure (see Fig. 14). By using shell elements, natural frequencies and mode shapes of this periodic structure are found for free boundary conditions. 29 modes are calculated up to 264.6 Hz. Yet, the next mode appears at 829.5 Hz. Therefore, the first stop band of this periodic structure is generated between these two frequencies. The two modes that set the limit for the stop band can be seen in Figs. 15 and 16. Notice that, in Fig. 15 the building block mechanisms deform similar to the first mode shape given in Fig. 10(a). Moreover, in Fig. 16 these mechanisms deform similar to their second mode shape as shown
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4.5
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ωp2/ω
p1
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Band depth (BD) Fig. 13. ωp2 =ωp1 ratio versus band depth of optimized mechanism designs. Red cross: selected size optimized design (BD ¼ 0.20, ωp2 =ωp1 ¼2.79) and blue circle: selected shape optimized design (BD ¼ 0.25, ωp2 =ωp1 ¼ 3.32). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 14. Two-dimensional phononic band gap structure.
Fig. 15. 29th mode shape of the phononic band gap structure (264.5 Hz).
Fig. 16. 30th mode shape of the phononic band gap structure (829.5 Hz).
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Fig. 17. Manufactured phononic band gap structure.
Fig. 18. Experimental setup.
3
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(ω)/y(ω)|
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Transmissibility,
:
1
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:
0
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Frequency (Hz) Fig. 19. Experimental and numerical acceleration transmissibility plots of the phononic band gap structure for the longitudinal direction (x1).
in Fig. 10(b). Therefore, the first two modes of the building block mechanism define the limits of the first stop band for the two-dimensional periodic structure. The out-of-plane thickness of the structure does not affect the frequency response for in-plane excitations. Therefore, the stop band frequency range is not a function of out-of-plane thickness of the structure. However, the out-of-plane bending modes of the structure are affected by this thickness. Numerical modal analysis results show that 55 mm is the minimum thickness that places the first out-of-plane bending mode above the stop band frequency range. In order to have a compact specimen that can be accurately manufactured, the out-of-plane thickness is selected as 55 mm. The periodic structure shown in Fig. 14 is manufactured via wire-EDM method (see Fig. 17). The material properties of the sample are the same with the numerical model (E ¼205 GPa, ρ ¼ 7800 kg/m3, ν ¼ 0:29).
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|x (ω)/y(ω)| 1
..
..
|x2(ω)/y(ω)|
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10 x (ω)
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x (ω) y(ω)
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Fig. 20. Experimental acceleration transmissibility plots of the phononic band gap structure for longitudinal (x1) and transverse (x2) directions.
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.. .. .. .. |x (ω)/y(ω)| |x1(ω)/y(ω)|
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Transmissibility
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Fig. 21. Experimental acceleration transmissibility plots of the phononic band gap structure for longitudinal (x1) and out-of-plane (x3) directions.
In the experimental study, the shape optimized periodic steel structure that is supported by rubber cords is excited from one end via a modal shaker (see Fig. 18). The input acceleration is measured by an impedance head and the output acceleration is measured by an accelerometer. As the noise level of the impedance head is smaller than that of the output accelerometer, H1 estimate [35] of the acceleration transmissibility frequency response function is used. The average of 90 measurements is calculated to reduce the effect of noise in the system. Moreover, Hanning window is utilized to reduce spectral leakage. In Fig. 19, finite element analysis and experimental frequency response results in longitudinal direction are compared. Here, the input acceleration is y€ and the output acceleration is x€1 . It can be seen that experimental and finite elements analysis results are consistent with each other and the stop band forms in the same frequency range (265–830 Hz). This graph also shows that, within the stop band, less than 0.5 percent of input vibration is transmitted to the output end. In Fig. 20, experimental results for transmissibilities for longitudinal and transverse directions are compared. For this case, the in-plane input makes 451 with respect to the x1- and x2-axes, while accelerations in longitudinal (x1) and transverse (x2) directions are observed as outputs. Less than 1 percent of input vibration is transmitted to the output end for this case. Again stop band limits are the same as before (265–830 Hz). Therefore, Figs. 19 and 20 indicate that the twodimensional periodic structure, which is formed by using the shape optimized building block mechanisms, provides a wide (between 265 and 830 Hz) and deep (BD 4 0:99) stop band. The experimental transmissibility results for the out-of-plane direction are given in Fig. 21. For this case, a longitudinal input with a slight out-of-plane inclination is fed to the structure while the accelerations in the out-of-plane (x3) and
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longitudinal (x1) directions are observed as outputs. It can be seen that by choosing 55 mm thickness for the steel sample, out-of-plane modes of vibration are avoided in the stop band frequency range. 6. Conclusion In this paper, a two-dimensional periodic structure, which has a wide and deep stop band at low frequencies is designed via inertial amplification method. Size and shape optimizations are performed on the one-dimensional beam element model of the building block mechanism to achieve wide and deep stop bands. For the same stiffness and mass values, the shape optimized design provides a wider and deeper stop band compared to the size optimized one. Besides, the relationship between band width and band depth is shown in Pareto studies in which shape optimization procedure provides better designs than that can be obtained by size optimization. Moreover, a shell element model of the building block mechanism is formed and this model shows similar stop band characteristics as the beam element model, indicating that the onedimensional finite element model is valid for a fast optimization procedure. Using the shell element model, a twodimensional periodic structure is formed. It is shown that the stop band limits of this periodic structure are nearly the same with its building block mechanism. Moreover, the two-dimensional phononic band gap structure is manufactured with a suitable out-of-plane thickness to prevent any out-of-plane vibration modes to occur within the stop band. Frequency response of the phononic band gap structure is obtained numerically and experimentally and it is found that they are close to each other. Experimental results show that, a wide and deep phononic gap is created for in-plane excitations in both longitudinal and transverse directions. It is also shown via experiments that out-of-plane vibration modes do not occur within the phononic gap due to the proper selection of the out-of-plane thickness of the periodic structure.
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