- Email: [email protected]

S0022-460X(18)30681-3

DOI:

10.1016/j.jsv.2018.10.014

Reference:

YJSVI 14430

To appear in:

Journal of Sound and Vibration

Received Date: 9 May 2018 Revised Date:

8 September 2018

Accepted Date: 8 October 2018

Please cite this article as: A.H. Orta, C. Yilmaz, Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism, Journal of Sound and Vibration (2018), doi: https://doi.org/10.1016/j.jsv.2018.10.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Adil Han Ortaa , Cetin Yilmaza,∗ a

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Department of Mechanical Engineering, Bogazici University, 34342, Bebek, Istanbul, TURKEY

Abstract

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Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism

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Keywords: phononic band gap, inertial amplification, flexures, chiral metamaterials

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1. Introduction

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Phononic band gaps are investigated in a one-dimensional (1D) array of compliant axial to rotary motion conversion mechanisms. To create low frequency band gaps, the effective inertia of the unit cell mechanism is amplified. It is shown that 1D array of this chiral unit cell generates a stop band limited by two different types of modes of the unit cell. The lower limit is governed by the fundamental coupled axial-torsional mode and the upper limit is governed by the bending mode. To maximize the isolation bandwidth, cross flexures and spiral flexures are utilized in the unit cell, which in turn provide high bending stiffness and low stiffness for coupled axial-torsional motions. Besides, helical steel wires with finite bending stiffness and large pitch angle are used to generate large rotational motion for axial excitations. Hence, the effective inertia of the unit cell is increased. As the axial and torsional stiffnesses of the system are low, and its effective inertia is amplified, a stop band occurs at low frequencies. Phononic band structure and the frequency response characteristics of the system are obtained by using analytical and finite element models. Parametric studies are conducted to create wide band gaps at low frequencies. Prototypes of the unit cell and the periodic structure are manufactured via 3D printing and laser cutting. Then, the analytical and computational frequency response results are compared with the experimental results for validation. In the end, a very wide low frequency stop band is realized.

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Periodic structures are studied considering their wave propagation characteristics in the literature [1, 2]. Infinite periodic structures can prevent wave propagation in certain frequency ranges. Frequency ranges in which acoustic or elastic waves do not propagate are called phononic band gaps [3, 4]. In contrast to infinite periodic structures, some amount of wave or vibration transmission is observed in finite periodic structures even if they are ∗

Corresponding author. Tel: +90 212 359 6436 Email addresses: [email protected] (Adil Han Orta), [email protected] (Cetin Yilmaz) Preprint submitted to Journal of Sound and Vibration

October 8, 2018

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excited within band gaps. This wave or vibration transmission can be found by the examination of frequency response function diagrams. Finite periodic phononic band gap structures can be used for filtering mechanical vibrations, and utilized in noise and vibration isolation systems, and acoustic or elastic wave guides. For this reason, the maximization of isolation bandwidths and isolation effectiveness within the bandwidth are very important for practical applications. There are two widely used methods to create phononic band gaps in a periodic medium. These are Bragg scattering and local resonances. Bragg scattering is commonly used to obtain phononic band gaps for acoustic and elastic waves [2, 5, 6]. In this method, wave speed and lattice parameter determine the lowest phononic band gap. Consequently, materials that have high density/low elastic modulus or large sized periodic structures have to be chosen to obtain band gaps at low frequencies. However, these requirements create practical challenges for low frequency applications. Local resonance is another method to create phononic band gap in a structure. It can create band gaps at much lower frequencies than Bragg scattering [7, 8, 9]. Rubber coated dense metals in an epoxy matrix can be used to create local resonances at low frequencies [7]. In this method, the gap size and the gap center frequencies are dependent on volume filling fraction, and independent of the geometric arrangement of the local resonators [10]. However, large volume fractions require larger mass because the density of the coated inclusions are higher than the matrix. Realizing heavy resonators in small size structures is a hard design problem. Recently, apart from Bragg scattering and local resonances an alternative method based on inertial amplification has been found [11, 12]. With inertial amplification, band gaps can be formed below the Bragg limit [11]. Moreover, inertial amplification induced gaps are shown to be qualitatively different from Bragg gaps or local resonance induced gaps in terms of wave energy localization characteristics or gap depth profiles in frequency response function plots [12]. Embedded inertial amplification mechanisms are investigated in 2D and 3D lumped parameter structures [11, 12, 13], and 1D, 2D and 3D distributed parameter structures [14, 15, 16, 17] to obtain wide phononic band gaps. Moreover, inertial amplification mechanisms can be attached to a host structure to obtain low frequency band gaps [18, 19, 20, 21]. These studies show that with the inertial amplification method, the disadvantages of Bragg scattering and local resonators can be eluded. Lower frequency limit of a band gap can be decreased by only increasing the amplification ratio without changing the mass or the stiffness of the system. An alternative approach for the use of inertial amplification mechanisms exploit axial-rotary motion coupling [22, 23]. However, realizing wide band gaps at low frequencies for these type of chiral structures is a hard task due to the presence of various types of resonances (bending, axial, torsional) of the unit cell mechanisms. In order to have a wide band gap, all types of resonances should be well separated. In this study, inertial amplification is used to obtain a wide band gap by using a novel axial to rotary motion conversion mechanism that includes cross flexures and parallel spiral flexures. The created wide band gap is shown through analytical, computational and experimental models.

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2.1. Lumped Parameter Models In traditional mechanisms, links and hinges are rigid [24, 25]. Due to relative motions, friction and backlash may be observed in these mechanisms. However, in a compliant mechanism, the links are connected by flexible hinges so that friction and backlash can be eliminated [26]. An example compliant mechanism is Linear-to-Angular-Displacement-Device (LADD) [27, 28] which can be seen in Fig. 1. If this type of a mechanism is considered for band gap generation, the lower limit of the band gap would be governed by axial-torsional mode of its unit cell and the upper limit of the band gap would be governed by the bending modes.

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The main goal of this study is to create wide and deep phononic band gaps at low frequencies by utilizing axial to rotary motion conversion mechanisms that show inertial amplification effect. To achieve this, a mechanism is designed to convert axial motion into rotational one. The main design considerations of the mechanism are to obtain high stiffness for bending and low stiffness for coupled axial-torsional motions. Thus, transverse modes of the mechanism will be prevented. Besides, effective inertia of the system will be increased to create wide band gaps at low frequencies.

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2. Models and Methodology

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Figure 1: Linear-to-Angular-Displacement-Device (LADD) mechanism used within the (a) unit cell and (b) 1D array. 88 89 90 91 92 93 94 95

LADD model shown in Fig. 1 has low bending stiffness, hence wide band gaps cannot be obtained. To increase the bending stiffness of the structure, a unit cell mechanism with two rings is proposed (Fig. 2 (a)). When this unit cell is used to form an array, the rings with mass m are constrained to translate axially whereas the rings with mass ma can translate axially and rotate. Due to these constraints, bending stiffness of the array will be high and wide band gaps will be produced. To calculate the band depth (attenuation) of the mechanism an analytical model is created based on lumped elements (Fig. 2(a)). In Fig. 2(a), when the base moves axially 3

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(assuming βi = 0), the upper ring with mass m only translates with displacement xi+1 whereas the lower ring with mass ma translates with displacement xi+1 and rotates with angular displacement βi+1 . Therefore, an axial input to the unit cell (xi ) generates axial (xi+1 ) and rotary (βi+1 ) motion.

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Figure 2: (a) The proposed unit cell mechanism. (b) 1D array of the proposed mechanism.

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To obtain the transmissibility and the phononic band structure, the bending stiffness of the wires has to be calculated. The wires are modelled as straight beams due to their high pitch angles. To find the stiffness, deflection and boundary conditions of the beam are modelled as shown in Fig. 3(a-b).

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Figure 3: Deflection of the wires (a) within the unit cell mechanism, (b) with simplified boundary conditions.

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Assuming small deflections, the bending stiffness of the wires [29, 30] are calculated as (Fig. 3(b)), F 12EIwire = kw = δ L3 Due to geometry (Fig. 3(a)), angular and axial displacements are related by, βi+1 − βi =

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Eq. (2) shows that the angular and axial motions of the disk with mass ma are coupled. Thus, in the unit cell mechanism, deflection of the wires create both rotational and translational motion (Fig. 3(a)). To calculate the potential energy of the unit cell, deflection (δ) can be taken as, (xi − xi+1 ) r(βi+1 − βi ) = (3) cos θ sin θ Considering that 3 wires are connecting the disks with mass m and ma , and assuming that xi as input displacement (βi = β˙ i = 0), potential (V ) and kinetic energy (T ) of the unit cell can be written as, δ=

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1 1 (xi − xi+1 )2 V = 3kw (δ)2 = 3kw 2 2 sin2 θ 1 1 2 T = (m + ma )x˙ 2i+1 + I β˙ i+1 2 2 When Eq. (2) is used in Eq. (5), T is obtained as,

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According to the potential energy equation (Eq. (4)), effective axial stiffness of the structure is equal to, 3kw (7) keff = sin2 θ The effective axial stiffness of the mechanism is compared with Ref. [31] that include finite element models, and the difference in the stiffness is around 10-15 percent. After the verification process, the Lagrange method [32] is utilized considering xi as input and xi+1 as output displacement, and equation of motion of the unit cell is obtained as, I cot2 θ I cot2 θ m + ma + x¨i+1 + keff xi+1 = x¨i + keff xi (8) r2 r2

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1 1 (x˙ i − x˙ i+1 )2 T = (m + ma )x˙ 2i+1 + I 2 2 2 r tan2 θ

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(xi − xi+1 ) r tan θ

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Notice that the term that multiplies x¨i+1 is the effective mass of the unit cell. If θ is small, effective mass of the unit cell can be much larger than m + ma , hence there is inertial 5

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At large frequencies (ω>>ωz1 ), transmissibility equation can be simplified as

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TR∞ =

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Here, γ represents the wave number. Notice that angular displacements do not appear in Eq. (13) as they are written in terms of axial displacements (see Eq. (2)). To obtain the phononic band structure, Eq. (13), Eq. (14), and Eq. (15) can be solved together. Thus, the dispersion relation can be obtained as, cos(γ) = 1 −

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The analytical model shows that TR∞ is independent of the stiffness keff . As the analytical model is SDOF, the effect of higher natural frequencies is neglected. However, Eq. (12) can be used for isolation band depth calculation. To obtain the phononic band structure of the 1D array in Fig. 2(b), equation of motion considering xi and its neighbors can be written by the Lagrange method, and the phononic band structure can be calculated by Bloch’s theorem.

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amplification. Moreover, the first resonance and antiresonance frequencies can be found respectively as s keff r2 ωp1 = (9) (m + ma )r2 + I cot2 θ r keff r2 (10) ωz1 = I cot2 θ Then, transmissibility of the structure can be calculated as 2 1 − ωω z1 TR(ω) = (11) 2 ω 1 − ωp1

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(m + ma )ω 2 r2 2(r2 keff − ω 2 I cot2 θ)

The phononic band structure plots will be examined in Section 3.

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2.2. Finite Element Models The analytical model of the unit cell (Fig. 2(a)) assumes that the ring with mass m translates axially and the ring with mass ma translates axially and rotates. Therefore, the rings m and ma should remain parallel to each other. The constraint that ensures this condition should provide low torsional stiffness but high bending stiffness. To that end, cross flexure mechanisms are considered [33, 34]. A cross flexure mechanism, which consists of two separate rings is designed, and these rings are connected with elastic beams (Fig. 4(a)). The slender beams in the system allow rotation under small torques. However, due to their large widths they prevent bending motion in which the two rings tend to separate from each other. In Fig. 4, an example cross flexure mechanism is depicted. The inner diameter of the rings is 100 mm, the overall height (2h) is 60 mm and the dimensions of the flexure beams are 80 mm×22 mm×0.2 mm. Notice that width of the flexure beam is 8 mm less than h=30 mm allowing 4 mm clearance on both sides. The flexures are made up of steel (E=200 GPa, ρ=7800 kg/m3 , µ=0.3) and the rings are made up of PLA (E=3.5 GPa, ρ=1190 kg/m3 , µ=0.36). Finite element model of the structure is solved with the non-linear geometry solver of the ABAQUS software, using 3D tetrahedral mesh. The first mode shape of the cross flexure mechanism is at 4.3 Hz in the torsional direction (Fig. 4(b)). The bending mode shape of the structure is at 104.0 Hz, which is much larger than the frequency of the torsional mode. If the width of the flexures is increased and their thickness is decreased, then the ratio of torsional and bending natural frequencies can be decreased further.

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Figure 4: Cross flexure mechanism. (a) CAD model. (b) First mode shape (torsional type at 4.3 Hz). (c) Second mode shape (bending type at 104.0 Hz). 161 162 163 164 165 166 167 168 169

The cross flexure mechanisms will be connected in series to form a periodic structure. The connections will transform axial motion into rotary motion. In the literature, spiral shaped flexure springs are commonly used to obtain high bending and low axial stiffness [35, 36, 37, 38, 39]. In each unit cell of the periodic structure, top and bottom surfaces of the upper ring will contain spiral flexures. 4 bolts near the center of the spiral flexures will connect individual unit cells to each other. Moreover, there should be room for the supports that are used to connect the rings inside the unit cells. According to these constraints, a steel spiral flexure is designed (Fig. 5(a)) and will be attached to the top and bottom surfaces of the upper ring. The main goal of the spiral structure of the flexures is to have 7

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high in-plane (radial) stiffness and low out-of-plane (axial) stiffness. To that end, the spiral arms are defined by splines and they are optimized to maximize radial to axial stiffness ratio. In the end, stiffness values turn out to be kx = 1130 N/mm, ky = 1240 N/mm, kz = 1.37 N/mm, giving kx /kz =824.8 and ky /kz =905.1. The spiral flexure in Fig. 5(a) will support the unit cell allowing axial motion while constraining bending. In order to see the dynamic characteristics of the spiral flexure, 4 cylinders having a total mass 0.5 kg are added to the central bolt holes and the first two mode shapes are computed (Fig. 5(b) and Fig. 5(c)). Notice that the first mode shape, which is in the axial direction, is at 10.8 Hz. The second mode shape that is in the bending direction is at 19.4 Hz. Hence, the second natural frequency is not far from the first one. In order to increase the bending to axial natural frequency ratio, two parallel spiral flexures with a spacing of 30 mm are used (Fig. 6). When two parallel spiral flexures are used, the mass of the system is doubled as well as its axial stiffness. Thus, the first natural frequency (11.2 Hz) is very close the single spiral flexure case (10.8 Hz). On the other hand, bending mode of the parallel flexures (185.2 Hz) is significantly higher than the single flexure case (19.4 Hz).

Figure 5: Spiral flexure. (a) CAD model in which the outer edge is fixed (outer diameter is 116 mm, thickness is 0.5 mm). (b) First mode shape (axial type at 10.8 Hz). (c) Second mode shape (bending type at 19.4 Hz).

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Figure 6: Two parallel spiral flexures. (a) CAD model in which the outer edge is fixed. (b) First mode shape (axial type at 11.2 Hz). (c) Second mode shape (bending type at 185.4 Hz).

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Cylindrical helical structures are commonly used to convert axial motion into rotational motion in the literature [27, 28, 40, 31]. These compliant mechanisms are ideal for small rotation cases due to their frictionless design. In Fig. 7, an example helical compliant mechanism is depicted in which the disks are 10 mm thick and their diameters are 100 mm. For symmetry, three 100 mm long wires with 2 mm radii are placed at 120 degrees around the disks. The helix angle of the wires are 60 degrees. The whole structure is made up of steel. When the bottom disk is fixed and roller boundary condition is applied to the cylindrical surface of the top disk, the first and the second natural frequencies are obtained at 122.2 Hz (Fig. 7(b)) and 827.4 Hz (Fig. 7(c)), respectively. Fig. 7(b) shows that as the top disk moves towards the bottom one, it rotates as well.

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In the final design, cross flexure mechanism, parallel spiral flexures and helical wires are integrated, and a compliant axial to rotary motion conversion mechanism is created (Fig. 8). In brief, basic working principle of the mechanism is based on transmitting the input axial displacement into the bottom ring of the mechanism (ma ) via helical wires. These wires create rotational movement for the bottom ring. At this point, parallel spiral flexures prevent the rotation of the upper ring (m) and increase the bending stiffness of the structure. Thus, the upper ring only vibrates axially. Moreover, cross flexures ensure that the rings with mass m and ma remain parallel to each other as ma rotates with respect to m. Plates and bolts are integrated above the upper ring to create a periodic structure. When Fig. 8(c) and Fig. 2(a) are compared, one can see that the proposed lumped parameter unit cell is realized as a compliant axial to rotary motion conversion mechanism. Side view of the unit cell can be seen in Fig. 9(a). Notice that there is a vibration clearance (marked by the double arrow) between consecutive M6 connection bolts. In Fig. 9(b), axial-torsional coupled mode shape is shown. Note that the central M6 connection bolts are attached to the parallel spiral flexures at the top and they are attached to the parallel connection plates at the bottom. As the top ring translates downward, the top spiral flexure deforms towards the vibration clearance (see Fig. 9(b)). Moreover, the middle ring both translates downward and rotates due to the presence of the helical wires. As a result, coupled axial-torsional motion is generated.

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Figure 7: Helical flexure hinges. (a) CAD model where the bottom disk is fixed and the upper disk has roller boundary condition at the cylindrical surface. (b) Axial-torsional coupled mode shape (122.2 Hz). (c) Bending mode shape (827.4 Hz).

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Figure 8: Final design of the unit cell. (a) Cross flexures within the unit cell. (b) Parallel spiral flexures within the unit cell. (c) Section view of the unit cell with helical wires, connection bolts and connection plates.

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Figure 9: Vibration of the unit cell. (a) Side view of the unit cell showing the vibration clearance (marked by the double arrow) between consecutive M6 connection bolts. (b) Axial-torsional coupled mode shape. Here, the top spiral flexure deforms towards the vibration clearance.

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3. Parametric Studies for Bandwidth Maximization In order to obtain maximum bandwidth, which will occur between the first two natural frequencies of the unit cell (w1 and w2 ), various parametric studies are conducted. The main goal is to minimize w1 /w2 ratio. However, complex geometry and connection properties of the structure require small finite element mesh, which increases model preparation and solution time. Since changing all design parameters at the same time is not feasible, they are examined separately. The inner diameters of the rings are taken as constant (100 mm). At the first part of the study, fundamental geometric parameters of the unit cell are examined 10

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on simplified models. In simplified models (Fig. 4, Fig. 7), the height of the rings (h) that also affect width of the flexure beams, thickness of the flexure beams (tbeam ), and the radii of the helical wires (rwire ) are examined, and the results can be seen in Tables 1-3. During the parametric study regarding the height of the rings (Fig. 4(a)), the width of the cross flexure beams are taken as 8 mm less than ring height (h) to create clearance between the spiral flexures and the cross flexure beams. The thickness and length of the cross flexures are taken as tbeam =0.2 mm and Lbeam =80 mm, respectively. Table 1 shows that an increase in the height of the ring increases torsional stiffness. However, the second natural frequency is the bending mode of the structure and increasing the thickness of the rings increases the mass of the structure. The parametric study about the thickness of the rings shows that w1 /w2 reaches minimum value when h=30 mm.

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Table 1: Natural frequencies and frequency ratio with respect to the ring height (h).

h (mm) w1 (Hz) 20 4.1 25 4.3 30 4.3 35 4.4 40 4.4

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For the selected ring height, further analysis is conducted for cross flexure beam thickness. Table 2 shows that increasing beam thickness increases the torsional stiffness. The smallest w1 /w2 ratio can be obtained at 0.2 mm. However, cross flexure mechanism carries steel connection plates to create a periodic structure as shown in Fig. 8(c). An estimated 0.2 kg mass is added at the top of the cross flexure mechanism to see how it affects the natural frequencies. The results in Table 2 show that frequency ratios of the cross flexure mechanism are the same with the massless case. However, natural frequencies of the mechanism significantly dropped. w1 /w2 ratios increase when tbeam increases. However, this beam will be laser cut and to prevent wrappage due to residual stresses, tbeam =0.3 is preferred.

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w2 (Hz) 91.9 103.3 104.0 104.4 105.0

Table 2: Natural frequencies and frequency ratio with respect to flexure beam thickness (tbeam ) with and without mass. Here, Lbeam =80 mm and wbeam =22 mm.

Without tbeam (mm) w1 (Hz) 0.2 4.3 0.25 5.8 0.3 7.3 0.35 9.2 0.4 11.2

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additional mass w2 (Hz) w1 /w2 104.0 0.041 111.8 0.052 114.5 0.063 112.5 0.082 105.6 0.106

With 0.2 kg mass w1 (Hz) w2 (Hz) w1 /w2 2.0 48.7 0.041 2.6 51.4 0.051 3.5 55.3 0.063 4.2 52.0 0.081 5.3 50.2 0.106

Next, the radius of the helical wires is considered. Table 3 shows that increasing the wire diameter, increases the natural frequencies considerably. Minimum frequency ratio is 11

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obtained for 0.5 mm radius. However, reducing the wire radius less than 0.5 mm does not improve w1 /w2 ratio much and for lower values the bending modes of the wires adversely affect the isolation bandwidth of the unit cell.

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Table 3: Natural frequencies and frequency ratio with respect to radii of the helical wires (rwire ).

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w1 /w2 0.082 0.084 0.087 0.094 0.148

The unit cell is created with 0.5 mm helical wire radius (rwire ), 30 mm ring height (h) and 0.3 mm flexure beam thickness (tbeam ). The connection plate thickness is taken as 1 mm. As seen in Fig. 8(c), there are two parallel connection plates above the ring with mass m. The distance between these plates (d) is taken as 7 mm. If one plate were used, then the bending modes would occur at lower frequencies. For two plates, if d were smaller, bending modes would decrease. d is taken as the highest possible value that allow proper clearance for vibration. Further parametric studies are conducted with the complete unit cell regarding connection bolt length (Lb ), and connection bolt diameter (dbolt ). The objective of the parametric studies is the minimization of w1 /w2 , which maximizes the isolation bandwidth. Another aim is to have high level of attenuation within the isolation bandwidth. Eq. (12) shows that transmissibility (TR∞ ) depends on (m + ma ), I, r, and θ. It is assumed that m = 2ma considering the mass of the connection plates and bolts. If I = ma r2 , Eq. (12) can be simplified to TR∞ =cot2 θ/(3+cot2 θ). When the target level of transmissibility is selected as 0.75, θ can be solved as 18◦ . The length of the helical wires changes with respect to helix angle and bolt length. The bolt length is selected as a parameter rather than the length of the helical wires due to the commercial bolt length constraints. Results of the parametric study of the unit cell is given in Tables 4-5. In Tables 4-5, the first, second and fourth natural frequencies and their ratios with respect to the first natural frequency are given for various connection bolt diameters and lengths. As mentioned before, optimization objective is to minimize the w1 /w2 but when the frequency response functions of the unit cell are examined, the second and the third natural frequencies do not affect the transmissibility due to their non-axial effects. However, an increase in the second natural frequency can increase the fourth one. So, both w1 /w2 and w1 /w4 are examined to see their effects. Table 4 shows that increasing connection bolt diameter slightly increases the first natural frequency because the head and nut parameters are changed. The bending natural frequency increase with respect to bolt diameter, as expected. The fourth natural frequency, which is axial-torsional coupled mode is also increased with increasing bolt diameter. However, the minimum w1 /w4 ratio, which affect transmissibility is obtained for 6 mm bolt diameter.

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w2 (Hz) 70.6 152.7 260.4 537.9 827.4

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rwire (mm) w1 (Hz) 0.5 5.8 0.75 12.9 1 22.7 1.5 50.5 2 122.2

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Table 4: Natural frequencies and frequency ratio with respect to connection bolt thickness (dbolt ). Here, Lbolt =90 mm and θ=18◦ .

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w1 /w2 0.240 0.233 0.221 0.215 0.207

w1 /w4 0.218 0.159 0.156 0.158 0.159

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w4 (Hz) 58.2 80.1 82.6 83.3 83.9

To determine the length of the connection bolts (Lb ) another parametric study is conducted. This parameter determines the length of the helical wires and the bending natural frequencies. The results are shared in Table 5. An increase in the bolt length decreases all natural frequencies and the frequency ratios. When a 1D array with multiple unit cells is considered, the upper limit of the stop band will be governed by the bending modes of the array. For large Lb values, the bending modes will occur at low frequencies. Consequently, Lb =90 mm is selected.

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w2 (Hz) 53.0 54.4 58.4 61.4 64.1

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dbolt (mm) w1 (Hz) 4 12.7 5 12.7 6 12.9 7 13.2 8 13.3

Table 5: Natural frequencies and frequency ratio with respect to length of the bolts. Here, dbolt =6 mm and θ=18◦ .

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Two different materials are used within the unit cell considering manufacturability of the design. PLA is used for the rings, and steel is used for flexures, plates, wires, nuts and bolts. In the unit cell mechanism, radius of the helical wires and inner diameter of the cross flexure mechanism are taken as 0.5 mm and 100 mm, respectively. Height of the rings are 30 mm and dimensions of the flexure beams are 80 mm x 22 mm x 0.3 mm. Helical wires have 72 degree helix angle (θ = 18◦ ), and their lengths are 80 mm. The helical wires are attached around the periphery of the rings yielding r = 61 mm. For the given variables, m + ma and I are calculated as 0.73 kg and 8.56×10−4 kg.m2 , respectively. The effective axial stiffness of the unit cell is calculated using Eq. (7) as (3kw / sin2 θ). However, as it can be seen from Fig. 8(c), two parallel spiral flexures are used, and their axial stiffness 2ks is calculated as 2 × 1.37 N/mm. Furthermore, torsional stiffness of the cross flexure beams can be obtained as,

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w1 /w4 0.189 0.172 0.156 0.151 0.143

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Lb (mm) w1 (Hz) w2 (Hz) w4 (Hz) w1 /w2 80 15.8 64.9 83.8 0.243 85 14.3 62.5 83.3 0.229 90 12.9 58.4 82.6 0.221 95 12.4 57.3 82.1 0.216 100 11.7 55.1 81.7 0.212

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Cross flexure beams are connected to rings with outer radius r. Helical wires are also at the outer radius of the rings, and convert axial motion into rotary one as shown in Fig. 3 and Eq. (3). Therefore, axial stiffness of the cross flexures can be calculated as, ktcross r2 tan θ In the end, effective axial stiffness of the unit cell in Fig. 8(c) is kcross =

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For the given parameter values, it can be calculated as 10.7 N/mm. When keff in Eq. (9) and Eq. (10) is replaced by kaxial in Eq. (19), wp1 and wz1 can be calculated as 9.7 Hz and 11.2 Hz, respectively. Hence, wp1 /wz1 ratio is equal to 0.87 giving TR∞ =0.75, which is the target transmissibility level in the isolation bandwidth (see Fig. 10). According to the finite element (FE) model, wp1 = 12.9 Hz and wz1 =14.6 Hz. Although there is a shift with respect to the analytical frequency values, almost the same wp1 /wz1 ratio of 0.88 is obtained in the FE study. Moreover, in the FE transmissibility result, there is a upper limit for the isolation bandwidth because of the higher frequency modes of the mechanism, which are not captured in the SDOF analytical model. Hence, transmissibility gradually increases towards the upper limit. Nevertheless, transmissibility level is less than 0.75 until w = 2wz1 , in the FE model.

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Figure 10: Comparison of analytical and finite element (FE) transmissibility plots for the unit cell mechanism. 311 312 313 314 315 316

Phononic band structure of the SDOF unit cell model is obtained by solving Eq. (16) for various helix angles (see Fig. 11). One can see that as the angle θ becomes smaller, the band gap starting frequency decreases. This is due to amplified inertia effect (see the multiplier of x¨i term in Eq. (13)). For θ = 18◦ , the band gap starts at 11 Hz. Moreover, there is a notch in the imaginary part of each plot due to the inertial coupling effect. The notch frequency decreases as θ decreases. For very low θ values, attenuation (band depth) 14

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above the notch frequency is small. To have considerable attenuation at low frequencies, θ is selected as 18◦ .

Figure 11: Complex band structure of the 1D lattice generated by the SDOF unit cell mechanism for θ = 5◦ , 18◦ and 30◦ .(a) Real wave vector. (b) Absolute value of the imaginary part of the complex wave vector.

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Mode shapes and natural frequencies of the unit cell can be seen in Table 6. The first, second and fourth mode shapes are depicted in Fig. 12.

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Figure 12: Mode shapes of the unit cell mechanism (a) First mode shape (axial-torsional coupled type at 12.9 Hz). (b) Second mode shape (bending type at 58.4 Hz). (c) Fourth mode shape (axial-torsional coupled type at 82.6 Hz). Table 6: Natural frequencies and mode shapes of the unit cell mechanism. In the last row, mode shapes have been represented as coupled axial-torsional (AT) or bending (B).

Mode Number Frequency (Hz) Mode Type

1 2 12.9 58.4 AT B

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Mode shapes and natural frequencies of the 1D periodic structure, which consists of 4 unit cells can be seen in Table 7. The first, sixth, seventh and eighth mode shapes are shown in Fig. 13.

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Figure 13: Mode shapes of the 1D array with 4 unit cells (a) First mode shape (bending type at 5.6 Hz). (b) Sixth mode shape (axial-torsional coupled type at 10.7 Hz). (c) Seventh mode shape (axial-torsional coupled type at 25.5 Hz). (d) Eighth mode shape (bending type at 39.2 Hz).

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In Fig. 14, transmissibility plots of arrays of the designed mechanism for 1, 2, 3 and 4 unit cells are shown. Frequency intervals in which transmissibility values are less than one give the stop bands. For all the plots, the first stop band appears approximately in 15-40 Hz. For lower number of unit cells, bandwidth is slightly larger. However, band depth, which defines the attenuation within the stop band, is larger for higher number of unit cells. For instance, the periodic structure with 4 unit cells provides at most 10% vibration transmission within 17-39 Hz whereas the periodic structure with 3 unit cells provides at most 30% vibration transmission within the same frequency range. In order to have considerable band depth, i.e., high amount of vibration attenuation, 4 unit cells are used within the prototype.

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Figure 14: Transmissibility plots of arrays of the designed mechanism for 1, 2, 3 and 4 unit cells.

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The optimized unit cell mechanism, which is depicted in Fig. 8(c) is used to create a 1D array with 4 unit cells (see Fig. 15(a-b)). Parallel spiral springs and connection plates are manufactured via laser cutting method, and unit cell rings are manufactured with 3D printing method. To connect the unit cell to the plates, M4 60 mm bolts and M6 90 mm connection bolts are used. To connect the helical wires to the PLA rings, snap rings are used. However, to have small contact area and provide a firm boundary condition, M3 slot head screws are inserted into the PLA rings, and the wires fit inside the slots (Fig. 15(c)). The overall size, mass and axial stiffness of the prototype can be seen in Table 8.

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Table 8: Size, mass and axial stiffness of the prototype.

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Total Length (mm) 510

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In the experimental study, the unit cell structure and the 1D array with 4 unit cells are tested. These structures are hung by rubber cords and excited from one end via modal shaker Fig. 16(a)-16(b). An impedance head measures the input acceleration, and an accelerometer measures the output acceleration. To reduce the effect of noise in the system, the average of 60 measurements are taken.

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Figure 15: The periodic structure with 4 unit cells. (a) CAD model. (b) Prototype. (c) Close up view of the wire boundary condition.

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Transmissibility of the unit cell can be seen in Fig. 17. Experimental results show that transmissibility is less than one between 14 Hz and 48 Hz. The ratio of the stop band limits is 3.4, which is quite large. Moreover, the resonance frequencies in the experiment and the finite element analysis (FEA) are almost the same. Notice that the second and third modes (see Fig. 12 and Table 6) did not appear in the transmissibility plot as they are bending modes that are not excited by an axial input. In general, there is a very good correlation between experimental and computational results.

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Figure 16: Experimental setup of (a) one unit cell, (b) periodic structure with 4 unit cells.

Figure 17: Comparison of experimental and finite element (FE) transmissibility plots for the unit cell mechanism. 354 355 356 357 358 359 360

Transmissibility of the periodic structure can be seen in Fig. 18. Experimental results show that transmissibility is less than one between 11 Hz and 39 Hz. The ratio of the stop band limits is 3.5, which is quite large. According to Table 7, the stop band is obtained between the sixth and eight modes (see Fig. 13). As the structure is excited from the bottom and the output is measured at the center of the top surface the seventh mode does not appear as a resonance peak. Due to zero displacement at the top center point (see Fig. 13(c)), it appears as an antiresonance notch at 25 Hz (see Fig. 17). The upper limit of 19

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the stop band is the eighth mode, which is of bending type. Maximization of the bending stiffness of the unit cell yielded a wide and deep stop band. Again, there is a good correlation between experimental and finite element results. The differences are originated from bolt prestresses, snap ring prestresses and assembly errors.

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A compliant axial to rotary motion conversion mechanism is designed to generate a wide band gap at low frequencies. There are two rings within the unit cell that are connected by cross flexures, which provide low torsional and high bending stiffness between the rings. The top rings of the unit cells are connected to consequent top rings through parallel spiral flexures, which yield low axial and high bending stiffness between the top rings. Lastly, there are helical wires with large pitch angle between the top and bottom rings of the consequent unit cells. Hence, axial motion between the top rings of the consequent unit cells generate large rotational motion for the bottom rings, which increase the effective inertia of the system. As the axial and torsional stiffnesses of the system are low, and its effective inertia is amplified, a band gap occurs at low frequencies. The lower limit of the band gap is governed by the coupled axial-torsional mode of the unit cell, which is its fundamental mode. Moreover, the cross flexures and the parallel spiral flexures increase the bending stiffness of the unit cell so that the upper limit of the band gap is attained at high frequencies. Lumped parameter and finite element models of the unit cell and a 1D array of 4 unit cells are generated, and used in parametric studies for isolation bandwidth maximization. Prototypes of the unit cell and the 1D array are manufactured and tested. Experimental and computational frequency response results match well and show that both the unit cell and the 1D array of the 4 unit cells have wide stop bands at low frequencies. In fact, the ratios of the upper and lower limits of the stop bands are more than 3, which is quite large.

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[1] L. Brillouin, Wave propagation in periodic structures: electric filters and crystal lattices, Dover, New York, 2003. [2] M. S. Kushwaha, Classical band structure of periodic elastic composites, International Journal of Modern Physics B 10 (09) (1996) 977–1094. [3] M. M. Sigalas, E. N. Economou, Elastic and acoustic wave band structure, Journal of Sound Vibration 158 (1992) 377–382. [4] J. V. S´anchez-P´erez, D. Caballero, R. M´artinez-Sala, C. Rubio, J. S´anchez-Dehesa, F. Meseguer, J. Llinares, F. G´ alvez, Sound attenuation by a two-dimensional array of rigid cylinders, Physical Review Letters 80 (24) (1998) 5325. [5] M. Kushwaha, B. Djafari-Rouhani, L. Dobrzynski, Sound isolation from cubic arrays of air bubbles in water, Physics Letters A 248 (2) (1998) 252–256. [6] Z. Liu, C. T. Chan, P. Sheng, Three-component elastic wave band-gap material, Physical Review B 65 (16) (2002) 165116. [7] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. T. Chan, P. Sheng, Locally resonant sonic materials, Science 289 (5485) (2000) 1734–1736. [8] M. Hirsekorn, P. P. Delsanto, N. Batra, P. Matic, Modelling and simulation of acoustic wave propagation in locally resonant sonic materials, Ultrasonics 42 (1-9) (2004) 231–235. [9] C. Goffaux, J. S´ anchez-Dehesa, Two-dimensional phononic crystals studied using a variational method: Application to lattices of locally resonant materials, Physical Review B 67 (14) (2003) 144301. [10] G. Wang, X. Wen, J. Wen, L. Shao, Y. Liu, Two-dimensional locally resonant phononic crystals with binary structures, Physical Review Letters 93 (15) (2004) 154302. [11] C. Yilmaz, G. Hulbert, N. Kikuchi, Phononic band gaps induced by inertial amplification in periodic media, Physical Review B 76 (5) (2007) 054309. [12] C. Yilmaz, G. Hulbert, Theory of phononic gaps induced by inertial amplification in finite structures, Physics Letters A 374 (34) (2010) 3576–3584. [13] S. Taniker, C. Yilmaz, Phononic gaps induced by inertial amplification in bcc and fcc lattices, Physics Letters A 377 (31-33) (2013) 1930–1936. [14] S. Taniker, C. Yilmaz, Generating ultra wide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges, International Journal of Solids and Structures 106 (2017) 129–138. [15] 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, Journal of Sound and Vibration 332 (24) (2013) 6389–6404. [16] O. Yuksel, C. Yilmaz, Shape optimization of phononic band gap structures incorporating inertial amplification mechanisms, Journal of Sound and Vibration 355 (2015) 232–245. [17] S. Taniker, C. Yilmaz, Design, analysis and experimental investigation of three-dimensional structures with inertial amplification induced vibration stop bands, International Journal of Solids and Structures 72 (2015) 88–97. [18] N. M. Frandsen, O. R. Bilal, J. S. Jensen, M. I. Hussein, Inertial amplification of continuous structures: Large band gaps from small masses, Journal of Applied Physics 119 (12) (2016) 124902. [19] J. Li, S. Li, Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects, Physics Letters A 382 (5) (2018) 241–247.

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This work was supported by Bogazici University Research Fund with Grant Number 16A06P4. Cetin Yilmaz acknowledges the support from the Turkish Academy of Sciences Distinguished Young Scientist Award (TUBA-GEBIP).

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Furthermore, the 1D array of the 4 unit cells provide a quite deep stop band, i.e., it achieves high level of vibration isolation in the targeted frequency range.

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[20] M. Barys, J. S. Jensen, N. M. Frandsen, Efficient attenuation of beam vibrations by inertial amplification, European Journal of Mechanics-A/Solids 71 (2018) 245 – 257. [21] J. U. Schmied, C. Sugino, A. Bergamini, P. Ermanni, M. Ruzzene, A. Erturk, Toward structurally integrated locally resonant metamaterials for vibration attenuation, in: Active and Passive Smart Structures and Integrated Systems 2017, Portland, Vol. 10164, International Society for Optics and Photonics, 2017, p. 1016413. [22] T. Delpero, G. Hannema, B. Van Damme, S. Schoenwald, A. Zemp, A. Bergamini, Inertia amplification in phononic crystals for low frequency bandgaps, ECCOMAS Thematic Conference on Smart Structures and Materials, Madrid, Spain, 2017, pp. 1657–1668. [23] B. Zheng, J. Xu, Mechanical logic switches based on dna-inspired acoustic metamaterials with ultrabroad low-frequency band gaps, Journal of Physics D: Applied Physics 50 (46) (2017) 465601. [24] A. Ugural, Mechanical design: an integrated approach, McGraw-Hill, New York, 2003. [25] M. F. Spotts, Design of machine elements, Pearson Education, India, 1971. [26] L. L. Howell, Compliant mechanisms, John Wiley & Sons, New York, 2001. [27] K. Pottebaum, J. Beaman, A dynamic model of a concentric ladd actuator, Journal of Dynamic Systems, Measurement, and Control 105 (3) (1983) 157–164. [28] G. Mennitto, M. Buehler, Experimental validation of compliance models for ladd transmission kinematics, in: Proceedings 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems. Human Robot Interaction and Cooperative Robots, Pittsburgh, Vol. 1, IEEE, 1995, pp. 385–390. [29] T. Yokoyama, Vibrations of a hanging timoshenko beam under gravity, Journal of Sound and Vibration 141 (2) (1990) 245–258. [30] J. S. Przemieniecki, Theory of matrix structural analysis, Dover, New York, 1985. [31] R. Wang, X. Zhou, Z. Zhu, Q. Liu, Compliant linear-rotation motion transduction element based on novel spatial helical flexure hinge, Mechanism and Machine Theory 92 (2015) 330–337. [32] D. J. Inman, Engineering vibration, third ed., Prentice Hall, New Jersey, 2008. [33] N. Lobontiu, Compliant mechanisms: design of flexure hinges, CRC Press, Boca Raton, 2002. [34] B. P. Trease, Y.-M. Moon, S. Kota, Design of large-displacement compliant joints, Journal of Mechanical Design 127 (4) (2005) 788–798. [35] N. Chen, X. Chen, Y. Wu, C. Yang, L. Xu, Spiral profile design and parameter analysis of flexure spring, Cryogenics 46 (6) (2006) 409–419. [36] E. Ozkaya, C. Yilmaz, Effect of eddy current damping on phononic band gaps generated by locally resonant periodic structures, Journal of Sound and Vibration 389 (2017) 250–265. [37] W. Zhou, L. Wang, Z. Gan, R. Wang, L. Qiu, J. Pfotenhauer, The performance comparison of oxford and triangle flexure bearings, in: AIP Conference Proceedings, Washington, Vol. 1434, 2012, pp. 1149– 1156. [38] M. Khot, B. Gawali, Finite element analysis and optimization of flexure bearing for linear motor compressor, Physics Procedia 67 (2015) 379–385. [39] R. VR, B. T. Kuzhiveli, Modelling and failure analysis of flexure springs for a stirling cryocooler, Journal of Engineering Science and Technology 12 (4) (2017) 888–897. [40] G. Mennitto, M. Buehler, Ladd transmissions: Design, manufacture, and new compliance models, Transactions-American Society of Mechanical Engineers Journal of Mechanical Design 119 (1997) 197– 203.

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Figure 1: Linear-to-Angular-Displacement-Device (LADD) mechanism used within the (a) unit cell and (b) 1D array.

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Figure 2: (a) The proposed unit cell mechanism. (b) 1D array of the proposed mechanism.

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Figure 3: Deflection of the wires (a) within the unit cell mechanism, (b) with simplified boundary conditions.

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Figure 4: Cross flexure mechanism. (a) CAD model. (b) First mode shape (torsional type at 4.3 Hz). (c) Second mode shape (bending type at 104.0 Hz).

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Figure 5: Spiral flexure. (a) CAD model in which the outer edge is fixed (outer diameter is 116 mm, thickness is 0.5 mm). (b) First mode shape (axial type at 10.8 Hz). (c) Second mode shape (bending type at 19.4 Hz).

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Figure 6: Two parallel spiral flexures. (a) CAD model in which the outer edge is fixed. (b) First mode shape (axial type at 11.2 Hz). (c) Second mode shape (bending type at 185.4 Hz).

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Figure 7: Helical flexure hinges. (a) CAD model where the bottom disk is fixed and the upper disk has roller boundary condition at the cylindrical surface. (b) Axial-torsional coupled mode shape (122.2 Hz). (c) Bending mode shape (827.4 Hz).

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Figure 8: Final design of the unit cell. (a) Cross flexures within the unit cell. (b) Parallel spiral flexures within the unit cell. (c) Section view of the unit cell with helical wires, connection bolts and connection plates.

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Figure 9: Vibration of the unit cell. (a) Side view of the unit cell showing the vibration clearance (marked by the double arrow) between consecutive M6 connection bolts. (b) Axial-torsional coupled mode shape. Here, the top spiral flexure deforms towards the vibration clearance.

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Figure 10: Comparison of analytical and finite element (FE) transmissibility plots for the unit cell mechanism.

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Figure 11: Complex band structure of the 1D lattice generated by the SDOF unit cell mechanism for θ = 5◦ , 18◦ and 30◦ .(a) Real wave vector. (b) Absolute value of the imaginary part of the complex wave vector.

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Figure 12: Mode shapes of the unit cell mechanism (a) First mode shape (axial-torsional coupled type at 12.9 Hz). (b) Second mode shape (bending type at 58.4 Hz). (c) Fourth mode shape (axial-torsional coupled type at 82.6 Hz).

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Figure 13: Mode shapes of the 1D array with 4 unit cells (a) First mode shape (bending type at 5.6 Hz). (b) Sixth mode shape (axial-torsional coupled type at 10.7 Hz). (c) Seventh mode shape (axial-torsional coupled type at 25.5 Hz). (d) Eighth mode shape (bending type at 39.2 Hz).

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Figure 14: Transmissibility plots of arrays of the designed mechanism for 1, 2, 3 and 4 unit cells.

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Figure 15: The periodic structure with 4 unit cells. (a) CAD model. (b) Prototype. (c) Close up view of the wire boundary condition.

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Figure 16: Experimental setup of (a) one unit cell, (b) periodic structure with 4 unit cells.

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Figure 17: Comparison of experimental and finite element (FE) transmissibility plots for the unit cell mechanism.

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Figure 18: Comparison of experimental and finite element (FE) transmissibility plots for the periodic structure with 4 unit cells.

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ACCEPTED MANUSCRIPT Highlights -A novel compliant axial to rotary motion conversion mechanism is designed. -The effective inertia of the designed unit cell mechanism is amplified. -Coupled axial-torsional and bending modes of the unit cell and the 1D array are analyzed.

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-Analytical, numerical and experimental frequency responses match.

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-Parametric studies are conducted to obtain wide and deep stop bands at low frequencies.

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