Measuring the band structures of periodic beams using the wave superposition method

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Measuring the band structures of periodic beams using the wave superposition method L. Junyi n, V. Ruffini, D. Balint Imperial College London, Mechanical Engineering Department, South Kensington Campus, London SW7 2AZ, United Kingdom

a r t i c l e i n f o

abstract

Article history: Received 16 September 2015 Received in revised form 30 June 2016 Accepted 4 July 2016 Handling Editor: G. Degrande

Phononic crystals and elastic metamaterials are artificially engineered periodic structures that have several interesting properties, such as negative effective stiffness in certain frequency ranges. An interesting property of phononic crystals and elastic metamaterials is the presence of band gaps, which are bands of frequencies where elastic waves cannot propagate. The presence of band gaps gives this class of materials the potential to be used as vibration isolators. In many studies, the band structures were used to evaluate the band gaps. The presence of band gaps in a finite structure is commonly validated by measuring the frequency response as there are no direct methods of measuring the band structures. In this study, an experiment was conducted to determine the band structure of one dimension phononic crystals with two wave modes, such as a bi-material beam, using the frequency response at only 6 points to validate the wave superposition method (WSM) introduced in a previous study. A bi-material beam and an aluminium beam with varying geometry were studied. The experiment was performed by hanging the beams freely, exciting one end of the beams, and measuring the acceleration at consecutive unit cells. The measured transfer function of the beams agrees with the analytical solutions but minor discrepancies. The band structure was then determined using WSM and the band structure of one set of the waves was found to agree well with the analytical solutions. The measurements taken for the other set of waves, which are the evanescent waves in the bimaterial beams, were inaccurate and noisy. The transfer functions at additional points of one of the beams were calculated from the measured band structure using WSM. The calculated transfer function agrees with the measured results except at the frequencies where the band structure was inaccurate. Lastly, a study of the potential sources of errors was also conducted using finite element modelling and the errors in the dispersion curve measured from the experiments were deduced to be a result of a combination of measurement noise, the different placement of the accelerometer with finite mass, and the torsional mode. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction Phononic crystals and elastic metamaterials are materials or structures that can be engineered to possess interesting properties that are not found in natural materials, such as negative effective stiffness, negative effective mass, and negative refraction [1–8], in certain frequency ranges. These unique properties give phononic crystals and elastic metamaterials the potential of being used

n

Corresponding author. E-mail address: [email protected] (L. Junyi).

http://dx.doi.org/10.1016/j.jsv.2016.07.005 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

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for a wide variety of applications, such as vibration and sound isolation [9–13], wave guiding [9,13], acoustic cloaking [13,14], and acoustic lensing [15,16]. Phononic crystals are periodic and although elastic metamaterials can be non-periodic, many analyses assumed periodicity when only analysing a single unit cell for elastic metamaterials. Thus, they are studied based on classical wave characteristics of periodic structures or crystals [6] and share some similar characteristics with photonic crystals. A notable example is the presence of band gaps, which are regions of frequencies where no wave can propagate, that exist in both photonic and phononic crystals [6]. Many designs to utilise the band gaps in this class of materials for vibration isolation have been proposed in the literature. Among the proposed designs include lattices with inertia amplification [12], chiral lattices with internal resonators [17], beams or plates with local resonators [18–20], structures with periodic piezoelectric patches [21–23], and many others. The band gaps are commonly determined by calculating the band structure, also known as the dispersion curve, of the material using a variety of methods, such as plane wave expansion (PWE) [24–27], extended plane wave expansion (EPWE) [28–31], finite element (FE) [32,33], transfer matrix (TM) [34,35], and spectral element (SE) [36–38] methods. Some studies, such as the ones performed by Airoldi and Ruzzene [23] and Bavencoffe et al. [39], have managed to measure the band structure from the response of a system by using laser velocimetry and then performing two-dimensional Fourier transforms. However, the method they used requires measurements with high temporal and spatial resolutions. It might also require post-processing of the data to remove the effects of wave reflection. Additionally, they did not manage to measure the band structure of the complex waves that characterises the attenuation within the band gaps. Junyi and Balint [40] have recently proposed an inverse method, known as the wave superposition method (WSM), to determine the band structure from a finite structure using a small number of data points. Their method allows the band structure to be calculated with simple equipment that are familiar to engineers, such as a shaker and accelerometers. This might encourage the adoption of elastic metamaterials and phononic crystals in engineering applications. Furthermore, WSM is capable of calculating both the real and imaginary parts of the band structure from a finite structure, which might be useful in the analysis of evanescent waves, as seen in [31]. In their paper [40], Junyi and Balint have only validated their method using analytical solutions of Timoshenko beams but in practice, the measured response of a system will contain noise and errors, for example, the noise in the accelerometer, manufacturing tolerances, and the accelerometers and force gauge having a finite area rather than a point. Therefore, the main objective of this study is to demonstrate that the technique is applicable in practice and discuss the issues while implementing WSM in a physical experiment. This is done by taking the required frequency response measurements of a beam and performing the calculations described in [40]. This paper is organised as follows. Firstly, the theory behind the wave superposition method from [40] will be briefly reviewed in the next section. Then, the experimental setup and properties of the beams studied will be described. In the next section, the results will be presented and discussed with emphasis on the issues related to this method. The relative displacement transfer function, which is the frequency response function of a point relative to that of a reference point, calculated from the measured band structure will be discussed. Finally, a study on the potential sources of error using finite element analysis is performed before concluding this paper.

2. Theory 2.1. Superposition of Bloch waves This section will briefly describe the basic principles of the wave superposition method and the calculation of the band structure using a finite number of points. An elastic wave in an infinite 1D periodic structure can be described by a Bloch

Fig. 1. Superposition of waves in a finite periodic structure.

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wave as shown in the following equation: ~ uðx þ L; tÞ ¼ uðxÞ eikL e  iωt

(1)

where u is the displacement, L is the lattice constant, u~ is a periodic wave function describing the displacements, x is the position, k is the wave vector, ω is the forcing frequency, and t is the time. Eq. (1) implies that the displacement at one end of a unit cell is a factor of eikL times the displacement at the other end of the unit cell. However, in a finite structure, a wave will be reflected when a discontinuity in geometry, material property, or loading condition is present. If the periodic structure is assumed to behave linearly, the waves can be superimposed, as described by Eq. (2), and the response at any point of the wave can be decomposed into two infinite periodic beams with waves of different magnitudes travelling in opposite directions as seen in Fig. 1: n1

Un ¼ λ

UF þ λ

1n

UR

(2)

where U n ¼ uðxÞn is the displacement at a given point in the n th unit cell, UF is the displacement of the forward moving wave at the first unit cell, UR is the displacement of the reverse moving wave at the first unit cell and λ is as defined in the following equation:

λ ¼ eikL

(3)

Additionally, for the bending of a beam studied in this paper, two pairs of opposite moving waves will be present and the response at any node is a sum of all four waves (two pairs of coupled waves) as described by the following equation: n1

W n ¼ λA

1n

W F; A þ λA

n1

W R; A þ λB

1n

W F; B þ λB

W R;B

(4)

where W is the transverse displacements and the subscripts A and B denote the individual waves. Closer inspection of Eq. (4) shows that for a finite beam, there are only a total of 6 unknowns: λA , λB , W F; A , W R; A , W F; A , and W R;A , if the frequency response is measured. Furthermore, each measured response contributes to an additional equation. Therefore, by measuring 6 points, there will be sufficient equations to solve for all the unknowns and determine the band structure for a beam in bending. Two equations, Eqs. (5) and (6), in terms of the response, λA and λB , can be obtained by substituting Eq. (4), for consecutive measurements and eliminating the magnitudes W F; A , W R; A , W F; A , and W R; A . Eqs. (5) and (6) can then be solved in order to determine the band structure of the beam. The solutions and further details for this analysis are described in [40]: ½W 3 λA  ðW 2 þW 4 ÞλA þ W 3 λA λB 2

4

þ½W 3 λA ð2W 2 þW 4 ÞλA þ ðW 1 þ 3W 3 ÞλA  ð2W 2 þ W 4 ÞλA þW 3 λB 4

þ ½  ðW 2 þ W 4 Þλ

3

2

λ

3

λ

λ

λ

4 3 2 2 A þðW 1 þ 3W 3 Þ A  ð3W 2 þ 2W 4  W 6 Þ A þðW 1 þ 3W 3 Þ A  ðW 2 þ W 4 Þ B

λ

λ

λ

λ

λ

4 3 2 ½W 3 A  ð2W 2 þ W 4 Þ A þ ðW 1 þ3W 3 Þ A ð2W 2 þW 4 Þ A þ W 3  B

λ

λ

λ ¼0

2 þ ½W 3 A  ðW 2 þW 4 Þ A þ W 3  A

(5)

½W 3 λA  ðW 2 þ W 4 ÞλA þW 3 λB 2

2

þ ½  ðW 2 þW 4 ÞλA þ ðW 1 þ 2W 3 þW 5 ÞλA  ðW 2 þ W 4 ÞλB 2

½W 3 λA ðW 2 þ W 4 ÞλA þ W 3 λA ¼ 0 2

(6)

2.2. Prediction of frequency response function from measured band structure In addition to determining the band structure, the wave superposition method also allows the prediction of the response at any other node using the measured band structure and the response at any four nodes. According to Eq. (4), the magnitudes of the individual Bloch waves within a finite structure for a given loading condition are constant regardless of the node number that the response is measured. It is also important to point out that λ and the wavenumber, k, are functions of the periodic structure and are not dependent on the loading conditions. Therefore, the response at any node can be determined by using the measured band structure and magnitudes. This can be easily done using the following equation: 2

3

2

1 λN1 A 6 7 6 N2 λA  1 6 W R;A 7 6 6 7¼6 6 6 W F;B 7 6 λN3  1 4 5 4 A 1 W R;A λN4 A

W F;A

1 λN1 A N2  1 λA 1 λN3 A N4  1 λA

1 λN1 B N2  1 λB 1 λN3 B N4  1 λB

3  12

1 λN1 B N2  1 7 7 λB 7 N3  1 7 7 λB 5 1 λN4 B

3 W N1 6 7 6 W N2 7 6 7 6 W N3 7 4 5 W N4

(7)

where the subscripts N1, N2, N3, and N4 denote the node numbers relative to a reference node. Once the magnitudes of the waves are found, the response at any node can then be calculated using Eq. (4). This principle can be expanded to enable the response of finite one-dimensional systems to be calculated with the band structure and the Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i

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eigenvectors for the systems as demonstrated in [40], which means simulations of finite systems using a full model are no longer necessary.

3. Experimental setup 3.1. Beam properties In this study, the band structures of the bending of two finite beams, one with a change in material properties and one with a change in geometry, were measured experimentally. The beam with a change in material properties is a bi-material beam (designated as Beam 1), while the beam with a change in geometry (designated as Beam 2) is a single material beam. Although the extension of the WSM to 2D and 3D structures has been discussed in the paper introducing the technique [40] and the technique can be applied with limitations to such structures, the beams here were selected to avoid many of the issues associated with the method with 2D and 3D structures. It is worth discussing some experimental difficulties with using the WSM in experimental situations. Firstly, as discussed in Section 5 of [40], the band structure of a 2D structure is a surface and the WSM method can only measure a portion of the surface. This measured portion is dependent on the boundary conditions. However, the boundary conditions cannot be easily controlled, especially for a 2D or 3D structure, for example the only boundary conditions that can be applied to a structure is loading at a point or a completely free boundary, while important boundary conditions such as equal displacements across a length cannot be applied in practice. Therefore, the WSM can only measure a curve that corresponds to the loading condition within the surface of the band structure of a 2D component instead of the entire surface. Apart from that, 2D and 3D structures have a larger number of evanescent waves as discussed in Section 5 of [40], and a way to alleviate this issue is to use a structure with a large number of unit cells and take the measurement of the unit cells in the middle of the structure, where the evanescent waves are negligible. However, this means that the tested structures are likely to be long, which may lead to many difficulties in setting up the experiment, for example, a long structure may deform due to its weight when it is hung freely in an experiment and may result in inaccurate measurements. Lastly, another difficulty in implementing the WSM for 2D and 3D structures is taking the required frequency response measurements for the structures, as the frequency response relating to the longitudinal waves of 2D and 3D structures cannot be measured with simple equipment like an accelerometer, reducing the utility of the WSM technique. Beam 1 was manufactured by using adhesives to attach beams of different materials together. Beam 2 was manufactured by milling a beam to the required dimensions. The material properties and dimensions of the two beams are listed in Table 1. Fig. 2 is an illustration of the dimensions of the beams used in Table 1. The damping in the material was modelled by using a complex stiffness as described by the following equation: E ¼ Eð1 þiηÞ

(8)

Table 1 Properties and geometry of the beams in the unit cell. Properties Beam 1 Material Young's modulus, E1 (GPa) Poisson's ratio, ν1 Shear modulus, G1 (GPa) Density, ρ1 (kg m  3) Loss coefficient, η1  10  3 Length, L1 (mm) Width, b2 (mm) Beam 2 Material Young's modulus, E2 (GPa) Poisson's ratio, ν2 Shear modulus, G2 (GPa) Density, ρ2 (kg m  3) Loss coefficient, η2  10  3 Length, L2 (mm) Width, b2 (mm) Others Thickness, h (mm) Unit cells, N Shear coefficient, κ

Beam 1

Beam 2

Poly-Carbonate 2.38 0.399 0.85 2710 16.80

Aluminium 70.00 0.330 27.00 2700 1.05

17 20

50 20

Aluminium 70.00 0.330 27.00 2700 1.05

Aluminium 70.00 0.330 27.00 2700 1.05

63 20

20 5

3 7 0.85

5 10 0.85

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Fig. 2. Illustration of beam dimensions.

Fig. 3. Dispersion curve for Beam 1. (a) Real part and (b) imaginary part.

Fig. 4. Dispersion curve for Beam 2. (a) Real part and (b) imaginary part.

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Fig. 5. Experimental setup. (a) Schematic and (b) photo of the experiment.

Fig. 6. Placement of accelerometer and stinger on beam.

The materials and dimensions of the beams were selected in order to produce band gaps at frequencies lower than 3000 Hz. The band gaps for Beam 1 and Beam 2 are 300–905 Hz and 1930–2230 Hz respectively, as shown in the analytical band structures plotted in Figs. 3 and 4. The dispersion curves were calculated based on the transfer matrix method with analytical Timoshenko beams as done in [40]. As seen in Figs. 3 and 4, the dispersion curves will have two pairs of waves corresponding to two modes of waves. One of the waves is purely evanescent, with only imaginary parts, while the other wave is propagating with only real parts with the exception of the region at the band gap. The purely evanescent wave shown in Figs. 3 and 4 is a result of the Timoshenko beam solution as seen in Eq. (9) and not due to material damping. According to Eq. (9), two of the exponents of e in the equation which corresponds to the wavenumber are real and two of them are imaginary. This results in one set of waves to be evanescent. The low amount of damping used in the models does not affect the band structure in this study, and the effects of damping on the band structure are discussed in [40]: h i Wðx; tÞ ¼ eiωt a1 e  ik1 x þ b1 eik1 x þ a2 e  k2 x þ b2 ek2 x (9) where W is the transverse displacement, k1 and k2 are the wavenumbers, and a1, b1, a2, and b2 are the constants that depend on the boundary conditions. 3.2. Experimental procedure The experimental setup is shown in Fig. 5. The beams were hung freely from the support frame using a fishing line. A stinger with a force gauge was then attached to the shaker that is also hanging freely from the support frame. The stinger with a load cell at the end was then attached to the beam at the first point of the first unit cell, as shown in Figs. 5 and 6, while the accelerometer was attached to the beam at the face opposite of the stinger. The accelerometer used in this study is a 352B10 ceramic shear ICP accelerometer by PCE piezotronics. The mass of the accelerometer is 0.7 g. The accelerometer, load cell and shaker were connected to a data acquisition unit to perform a frequency sweep using a sinusoidal signal. The frequency sweep was first done for the case where the accelerometer was placed right after the interface between the first and the second unit cell as shown in Fig. 6. The frequency range was 10–1200 Hz for Beam 1 and 50–3000 Hz for Beam 2. The frequency response (acceleration per unit force) was measured and recorded. The procedure was then repeated with the accelerometer placed at the next unit cell until the end of the beam. The measured frequency response was then used to calculate the band structure of the beams using WSM. Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i

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Fig. 7. Transfer function of polycarbonate and aluminium bi-material beam (Beam 1). (a) Point 2, (b) Point 3, (c) Point 4, (d) Point 5, and (e) Point 6. Analytical band gaps are highlighted in grey.

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Fig. 8. Transfer function of aluminium beam with different widths (Beam 2). (a) Point 2, (b) Point 3, (c) Point 4, (d) Point 5, (e) Point 6, (f) Point 7, (g) Point 8, and (h) Point 9. Analytical band gaps are highlighted in grey.

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4. Results and discussion 4.1. Frequency response function The measured and predicted frequency response for Beam 1 and Beam 2 are plotted in Figs. 7 and 8 respectively. The values in these plots were normalised with the frequency response at the first node. The normalised frequency response function will be referred to as the relative displacement transfer function, or transfer function herein. As seen in Fig. 7, the measured response has a very good agreement with the predictions using the analytical Timoshenko beam at all points with the exception of the responses around 350 Hz, in which an additional mode at 333 Hz is present. This error may be a result of a torsional mode that has a calculated natural frequency of 363.4 Hz, which was not taken into account while calculating the response as only the transverse response is of interest. However, the accelerometer may have picked up the effects of this mode. It is very difficult to place the shaker and accelerometer at the exact mid-point of the beam in practice, which makes this issue hard to eliminate. Alternatively, the error may also be caused by other bending modes that are not accounted for in the Timoshenko beam model or the moving of the accelerometer that have a finite mass during the experiment. The effects of these potential sources of error will be discussed in Section 5. Similarly, the transfer functions of Beam 2, as seen in Fig. 8, also agree well with the predictions made using the analytical Timoshenko beam, although there are still some discrepancies at several regions, for example, the transfer function at 250–395 Hz and additional modes of vibrations at 2280 and 2730 Hz. These modes may be caused by the torsional modes calculated to be at 2492.7 and 3036.4 Hz. Furthermore, the discrepancy between the analytical and experimental values around 2250–2500 Hz for the second node in Fig. 8(a) is large.

Fig. 9. Dispersion curve for Beam 1. (a) Wave A (real part), (b) Wave A (imaginary part), (c) Wave B (real part), and (d) Wave B (imaginary part). Analytical band gaps are highlighted in grey.

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Apart from that, the band gaps that are present in Beam 1 can be inferred from the transfer function in Fig. 7, where there are low transmissibilities in the band gap region of 300–905 Hz with the exception of the peak caused at 333 Hz that is likely to be caused by the torsional mode. Conversely, for Beam 2 the transmissibilities at the band gaps appear to be high and there are no obvious band gaps in the transfer function. This is due to the very low attenuation constant at the band gap, as seen in Fig. 4(b), of less than 0.1 and the magnitude of the wave is still large when it is reflected at the end of the finite beam. Apart from that, the visibility of the effects of the band gap on the transfer function depends on the width of the bandgap with respect to the modal density of the structure. For a narrow band gap in Beam 2, there is a possibility that the excited modes are the neighbouring modes, which leads to less reduction in the transfer function. Therefore, for systems that have band gaps with low attenuation constants and narrow band gaps, it is important to consider the band structure when demonstrating the presence of the band gaps experimentally. 4.2. Dispersion curves The dispersion curve for Beam 1 calculated using the measured transfer function and the wave superposition method is shown in Fig. 9. Since there are two set of waves, Wave A and Wave B, the measured points in the band structure were sorted so that one set of waves (Wave A) is shown in Fig. 9(a) and (b), while the measured points for the other waves (Wave B) are shown in Fig. 9(c) and (d) for clarity. As seen in Fig. 9(a) and (b), the measured points agree well with the analytical predictions of the band structure albeit the presence of some scatter. There are some discrepancies around the frequency of approximately 300 Hz in Fig. 9(a) and (b), which is likely caused by the errors in the transfer function measurement there.

Fig. 10. Dispersion curve for Beam 2 (combined). (a) Wave A (real part), (b) Wave A (imaginary part), (c) Wave B (real part), and (d) Wave B (imaginary part). Analytical band gaps are highlighted in grey.

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Another interesting observation that can be made in Fig. 9(b) is that the measured imaginary part of the wave vector at the band gap is fairly accurate with values close to the predictions made by the analytical solutions. This demonstrates that the wave superposition method is able to measure the imaginary part of the wave vector with adequate accuracy. This is a capability that is unique to the wave superposition method and has directly proved that the presence of that imaginary waves exists within the band gap. On the other hand, the results of Wave B for Beam 1, shown in Fig. 9(c) and (d), agree poorly with the predictions made using the analytical solutions and have a large amount of scatter. This is caused by the errors in the measurement in the second set of waves. As seen in Fig. 9(c) and (d), Wave B consists of evanescent waves in the beams, which attenuate across each unit cell. As seen in Fig. 9(c) and (d), the imaginary part of the wave vector of the evanescent waves (Wave B) is very large, which suggests that the attenuation of the evanescent waves are much larger than the propagating wave even at the band gap. This leads to the measurement of the band structure of the evanescent waves being very sensitive to errors in the measured response as the amplitude of these waves will be very low even after the first unit cell. Thus, any noise or ‘contamination’ by other modes will result in significant errors in the band structure calculations for the evanescent waves. As the evanescent waves attenuate very rapidly, they could be neglected when analysing phononic crystals or acoustic metamaterials, especially for vibration isolation applications, as the band gap that needs to be considered is that of the propagating waves (Wave A) in Fig. 9(a) and (b). Furthermore, some methods used to calculate the band structure by applying the Bloch wave boundary conditions and then finding the eigenvalues across the first Brillouin zone, such as the one in [41], are unable to obtain the band structure for the evanescent waves. Hence, in terms of analysing the location of band gaps for phononic crystals

Fig. 11. Dispersion curve for Beam 2 from points 2 to 7. (a) Wave A (real part), (b) Wave A (imaginary part), (c) Wave B (real part), and (d) Wave B (imaginary part). Analytical band gaps are highlighted in grey.

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and acoustic metamaterials, the evanescent waves can be omitted from the analysis. The measurement technique used in [23] was also unable to measure the band structure of the waves that correspond to Wave B. The errors found in the measured data also highlight the fact that the wave superposition method described here and in [40] has a key issue of not being able to reject measurement errors. As discussed in the previous section, the wave superposition method has equal numbers of equations and unknowns. Therefore, the system is fully determined and any error at any point of the measurements of the frequency response will result in large errors in the calculated band structure. Additionally, other factors, such as the size of the load cell and the accelerometer being of a finite radius instead of an infinitesimal point, the placement of the accelerometer not being exactly one unit cell length away, and manufacturing tolerances may result in minor differences in the experimental data compared to the analytical solution. Furthermore, the accelerometer in the experiment was moved to the point of measurement for the frequency response measurements at each point. The mass of the accelerometer may lead to the frequency response being different and introduce additional modes. Since the accelerometer is only applied at one point during each measurement, the measured beam is no longer periodic and since the WSM is developed based on the measured structure being periodic, the errors introduced by the addition of the mass may be significant. This issue may be alleviated by adding identical accelerometers for the entire beam or using contactless measurement methods, such as laser vibrometry, but these equipment were not available for this study. These issues might lead to large errors in the measured dispersion curve. Hence, a key improvement for the wave superposition method is to enable it to reject errors. A possible method to achieve this is to perform regression analysis. However, the equations to be solved for the WSM as shown in Eqs. (5) and (6) are complicated and involve complex numbers. Therefore, developing a scheme for the regression analysis for the WSM is not trivial and is a topic for future study. Since the frequency response function of Beam 2 was measured at 9 points, the dispersion curve can be calculated with the wave superposition method using any of the following 6 points: Points 1–6, Points 2–7, Points 3–8, and Points 4–9. The measured points of the band structure using all four sets of frequency response functions are plotted in Fig. 10. As seen in Fig. 10(a) and (b), the measured band structures of all four sets of data for Wave A are almost identical with the points overlapping each other. This should be the case as the band structure is a characteristic of the wave in the structure and should be identical throughout. Although the overall real parts of the wave vector in Fig. 10(a) are almost identical, there appear to be significant differences between the imaginary part of the wave vectors, especially near the band gap region as seen in Fig. 10(b), where the points do not overlap. Again, the likely cause of the errors near the band gap is the presence of the additional modes of vibration at 1904.8 and 2492.7 Hz that were not accounted for in the WSM and Timoshenko models. Despite the agreement for the dispersion curve of Wave A calculated using different datasets in Fig. 10(a) and (b), the dispersion curves of Wave B, as seen in Fig. 10(c) and (d), were found to be very noisy and the values calculated from each dataset differ significantly from each other. This is likely caused by the errors in the measured frequency response functions as explained previously. Fortunately, Wave B is always attenuating with large imaginary wave vectors and the wave attenuation property of the structure will be determined by the band gaps in Wave A. The dispersion curve with the dataset that gives the best result for Beam 2 (Points 2–7) when compared with the analytical solution is shown in Fig. 11. The observations made for the dispersion curves for Beam 2 are similar to that of the observations made for Beam 1. Firstly, for Wave A the experimental results agree well with the analytical solution as seen in Fig. 11(a) and (b). However, the measured band gap appears to be at the lower frequency than the predictions made by the analytical solution. The importance of being able to predict the band gap from the band structure is demonstrated here as the presence of the band gap is not obvious from the transfer function, shown in Fig. 8, due to the low values of the imaginary part of the wave vector for this structure. The dispersion curve of Wave B in Fig. 11(c) and (d) for Beam 2 does not

Fig. 12. Transfer function predictions of aluminium beam with different widths. (a) Point 8 and (b) Point 9. Analytical band gaps are highlighted in grey.

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agree well with the analytical predictions, similar to the case with Beam 1 shown in Fig. 9(c) and (d). The causes of the discrepancies of Wave B have been discussed previously. Lastly, based on the measured dispersion curves for both beams, it was observed that the accuracy for the propagating waves (Wave A) is relatively good, while the measured evanescent waves (Wave B) were found to be inaccurate. Therefore, the WSM is applicable in determining the propagating waves when experimental measurements are used. However, as mentioned previously, the band gap location is determined by the propagating waves and therefore, this technique can still be used as such to provide insight into characteristics of the propagating waves. 4.3. Transfer function predictions As discussed in Section 2.2, the measured band structure and the frequency response or transfer functions of 4 nodes can be used to predict the response of the other nodes of a finite structure using Eq. (7). In order to demonstrate this, the predictions of the transfer function of Beam 2 will be shown in this section. The best band structure was the one calculated using the transfer function from Points 2–7 (11). The measured band structure and the transfer function at Points 2–5 were used in Eq. (7) to determine the values of W F; A ; W R; A ; W F; B ; and W R; A for each excitation frequencies and then to calculate the transfer function of the beam at all of the nodes within the beam. Since the band structure was calculated using the transfer function of Points 2–7, the predicted transfer function using the measured band structure at these points will be identical to the measured values. Therefore, only the transfer function at Points 8 and 9 for the measured and predicted transfer function is compared in Fig. 12(a) and (b). The overall predicted transfer function using the measured band structure calculations shown in Fig. 12 (a) and (b) agrees well with the measured transfer function. In Fig. 12, the predicted transfer function for a large portion of the frequencies measured is similar to the measured transfer function especially for the transfer function at Point 8. The only significant discrepancy is the additional natural frequency at approximately 2500 Hz. Although the predicted transfer function for Point 9 agrees fairly well with the measured transfer function, the predictions are less accurate compared to that of Point 8. This is likely to be due to Point 9 being further away from the points in which the band structure was measured. There are some qualitative differences between the prediction and experimental values at frequencies 255–480 Hz and 1595–1640 Hz, while the additional mode at 2500 Hz is also present in the transfer function of Point 9. The errors around the frequencies 255–480 Hz and 1595–1640 Hz can be attributed to the errors in the measured transfer function, shown in Fig. 8, at these regions. Despite these errors, the method shown in this section appears to able to predict the transfer function of a finite system using the measured band structure with fair accuracy. However, it is worth pointing out that both Points 8 and 9 are relatively close to the other points where the measurements to make the predictions were taken. Since the accuracy of the WSM in predicting the frequency response is dependent on the accuracy of the measured band structure and the transfer function used to determine the band structure, it is crucial that the band structure measurements are accurate for accurate predictions of the transfer function for the WSM. Eq. (7) in Section 2.2 suggests that the predicted points of the response are dependent on the exponential of the product of the wave vector and the number of unit cells from the reference measurements. Therefore, any errors in the measured wave vectors and/or the reference frequency response functions will lead to errors in the predicted response and the quality of the predictions is likely to decrease for the points that are further from the measured points as the errors are amplified.

5. Effects of the potential sources of errors 5.1. Analysis setup Several potential sources of errors have been identified in the previous section. Therefore, a study has been performed using finite element analysis and the analytical Timoshenko beam model to investigate the effects of these sources of error on the measured dispersion curves using the wave superposition method. The potential sources of errors explored here are as follows:

   

Measurement noise. The excitation force being applied over an area instead of being a point force. The mass of the accelerometer. The torsional modes of the beam.

Beam 1, which is the bi-material beam made from aluminium and polycarbonate, as described in Section 3.1, was used to study the effects of these sources of error. In order to study the effects of measurement noise, noise with a uniform distribution ranging from 7 10 percent of the frequency response was added to the predictions made using the Timoshenko beam for the free–free beam loaded at one end, which is similar to the experimental conditions. A finite element model of the beam made of shell elements has been constructed to study the other sources of errors. This is because the finite Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i

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Fig. 13. Portion of the beam used to illustrate the finite element mesh and boundary conditions.

Fig. 14. Measurement points taken for error analysis. (Not to scale. Number of elements reduced for clarity).

element method allows other loading conditions, such as a force being applied over an area, to be studied. These loading conditions cannot be applied to the analytical Timoshenko beam model. In this study, the finite element model was constructed in ABAQUS using four node rectangular elements (S4R) and each element has a representative length of 1.25 mm, which means that there are 16 elements across the width of the beam and 448 elements across the entire length of the beam. The beam used for the finite element model is shown in Fig. 13, where the portion of the beam used to illustrate the meshes and loading conditions in Figs. 14 and 15 is highlighted. The measurements were taken at the centre node of the interface between each unit cell for all cases studied, except for the torsional modes, where the measurements were taken at the node 2.5 mm above the centre. The measurement points are shown in Fig. 14. The first loading condition that was studied is the uniform load at the side of the beam, shown as the green points in Fig. 15. This loading condition corresponds to the loading condition in the analytical Timoshenko Beam analysis. The second case is the area loading condition, where the nodes across a region, shown as the red points in Fig. 15, were loaded. This is similar to the loading condition in the unit cell where the load is applied over a region. Lastly, the effects of the torsion were studied by shifting the nodes of the area loading condition 2.5 mm above the centre, as shown by the blue dots in Fig. 15. For all loading conditions, an equal force in the out of plane direction is applied on all the loaded nodes and the sum of the forces was equal to 1 N. In order to study the effects of shifting the accelerometer, a point mass of 0.7 g, which is the mass of the accelerometer, was added to the measurement point of interest and the calculations performed on that point. The simulations were repeated for every single measurement point to simulate the experiment with the accelerometers being shifted. 5.2. Results and discussion The transfer functions for the analytical Timoshenko beam model with and without noise, and the plate finite element model with both the uniform load at the side and the area loading conditions are shown in Fig. 16. The transfer function for the plate with uniform load is similar to transfer function for the Timoshenko beam with slightly different natural frequencies for each mode. The magnitude of the transfer function at the natural frequencies for the finite element plate is higher than that for the Timoshenko beam as no damping was added to the finite element model because the finite element model in ABAQUS does not have hysteretic damping. The similarities are expected, as the plate model has the same loading condition as the Timoshenko beam. Overall, the minor differences between the finite element plate model with uniform load and the analytical solution can be attributed to a combination of the lack of damping, discretisation errors in the finite element model, and the simplification of the analytical Timoshenko beam model. On the other hand, the finite element plate model with the area loading condition has a fairly different transfer function at the lower frequencies, notably at frequencies lower than 200 Hz compared to the other loading conditions, as shown in Fig. 16 for all measured points. This result suggests that the loading conditions will affect the transfer function. Additionally, it is important to note that the transfer functions in Fig. 16 were calculated by dividing the amplitude of the point of interest to the amplitude of the first point. This leads to the slightly different resonance frequencies for the area load and uniform load conditions observed in the figure, even though the amplitudes for a given unit load are largest at the natural frequencies of the part. Although there are not many differences in the transfer function between the experimental transfer functions and the Timoshenko beam model for Beam 1 in Fig. 7, the loading conditions, where the force is excited over an area, may be a possible cause for the differences in the transfer functions between the experiments and analytical solutions at frequencies around 250–395 Hz for Beam 2, as seen in Fig. 8. Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i

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Fig. 15. Loading for studies (Not to scale. Number of elements reduced for clarity). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this paper.)

The dispersion curves calculated with the transfer functions for the different cases, in Fig. 16, using the wave superposition method are shown in Fig. 17. It was demonstrated in [40] that the dispersion curves calculated using the transfer matrix method and the WSM are identical for the transfer functions from the analytical Timoshenko beam. As seen in the figure, the dispersion curves for the Timoshenko beam without the noise and the finite element plate model with uniform load condition are very similar. This result is expected as the transfer functions of these two cases are almost identical. Despite the different transfer functions at low frequencies of the plate model with the area loading compared to the other loading conditions, the dispersion curves calculated using the WSM appear to be similar to the other loading conditions. This is because the waves in the area loading condition are still the sum of the two Bloch waves that correspond to the bending mode. The plate finite model can be viewed as a two-dimensional structure but for the uniform and area loading conditions, only the bending modes are excited. On the other hand, there is a large amount of scatter in the dispersion curve for the Timoshenko beam with noise. Despite the noise in the dispersion curve, the overall shape of the real and imaginary parts of Wave A of the case with the noise still follow that of the solution without the noise. Conversely, the general shape of the real and imaginary parts of Wave B of the case with the added noise no longer follow the trend for both the real and imaginary parts of the waves. This observation is similar to the results of experimental measurements for Beams 1 and 2, shown in Figs. 9 and 10 respectively. This suggests that noise in the transfer function is one of the main causes of the scatter and errors in the measured dispersion curves using the wave superposition method. Apart from noise, the effects of shifting the accelerometer position as done in the experiment and the effects of the torsional modes were also studied with the finite element model. Since the plate model with the area load gives accurate dispersion curves as seen in Fig. 17, this loading condition will be used as the reference for the study of these effects as this loading condition is closer to that of the experiments. The transfer functions for the cases with moving accelerometers and torsional modes are shown in Fig. 18. As seen in the figure, the transfer functions of the finite element plate with shifting accelerometer positions are similar to that for the Timoshenko beam but with additional modes and differences in the transfer function at lower frequencies, for example, the additional modes seen in Fig. 18(d). Similarly, the transfer functions for the case with the torsional modes are similar to that of the reference plate model but with two additional modes at around 372 and 741 Hz. Significant differences in the transfer function around this region were also observed. The measured experimental results are also shown in Fig. 18, and as seen in the figure the torsional modes do not appear to be measured in the transfer function. Furthermore, the additional mode around 333 Hz might be due to either the moving of the accelerometer or the measurement of the torsional modes at several locations. It must be noted that the accelerometers in the experiment are not likely to be placed at the exact locations as the finite element analysis. The dispersion curves calculated for the reference case and the case, in which the accelerometer was shifted, are shown in Fig. 19. As seen in Fig. 19(a) and (b), the dispersion curves with the moving accelerometer for Wave A are almost identical to that of the reference except for several regions at the lower frequencies and around 900–1000 Hz. However, for Wave B, the WSM can still calculate the real part accurately, as seen in Fig. 19(c), but the imaginary part in Fig. 19(d) was incorrectly calculated, where the curve is not accurate, although a distinct pattern exists near the band gap region. The dispersion curves calculated using the transfer functions of the torsional modes and the reference plate model are shown in Fig. 20. As seen in Fig. 20(a) and (b), the predicted transfer function for Wave A is fairly accurate for both the real and imaginary components at lower frequencies when the torsional modes are excited and measured. The main errors for Wave A appear to be around 400 Hz that is close to the predicted torsional modes of around 370 Hz. However, the torsional mode at 741 Hz does not appear to have been captured in Wave A. On the other hand, the prediction for Wave B appears to be inaccurate, as shown in Fig. 20(c) and (d), where the measured dispersion curves do not resemble the predicted values. Based on the observed effects studied in this section, the errors in the dispersion curves measured from the experiment, as seen in Figs. 9 and 10, are likely to be caused by a combination of noise, changing the placement of the accelerometer, and the torsional modes. The torsional modes should have the least impact on the results, as the transfer functions in the experiments for both beams, as seen in Figs. 7 and 8, do not appear to be significantly affected by them. However, despite the errors, the WSM is still capable of measuring the propagating wave (Wave A in this study) fairly accurately, which allows the prediction of the band gap. Unfortunately, the WSM is unable to measure the evanescent waves (Wave B in this study) accurately when other effects, such as noise, are present and therefore cannot be used to study the evanescent waves in such cases. Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i

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Fig. 16. Transfer function of polycarbonate and aluminium bi-material beam for error analysis (Beam 1). (a) Point 2, (b) Point 3, (c) Point 4, (d) Point 5, (e) Point 6, and (f) zoomed in portion of Point 2. Analytical band gaps are highlighted in grey.

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Fig. 17. Dispersion curve for Beam 1 for error analysis. (a) Wave A (real part), (b) Wave A (imaginary part), (c) Wave B (real part), and (d) Wave B (imaginary part). Analytical band gaps are highlighted in grey.

6. Conclusion A study on the band structure and band gaps of beams with variations in geometry and material properties was conducted. In this study, the wave superposition method introduced in [40] was used to determine the band structure of a onedimensional periodic structure with two waves from experimental measurements. The wave superposition method also allows predictions to be made for the response of other points in a finite periodic structure for a given boundary condition using information from the band structure and a few response measurements. The experiments were conducted to measure the band structure of two different beams, one consisting of 7 unit cells made from aluminium and polycarbonate sections and the other beam being an aluminium beam consisting of 9 unit cells with varying widths. The band structures of the beams studied were calculated using the wave superposition method [40]. The bi-material beam was constructed by attaching the individual sections using an epoxy based adhesive, while the beam with varying geometry was milled using a CNC machine. The beams were then hung freely from a frame using fishing threads and the beams are excited at one end using a shaker, while the acceleration at different unit-cells for a range of frequencies was measured using an accelerometer. The transmissibilities of the beams measured from the experiments were found to agree with the analytical solution. However, some discrepancies were observed and they were attributed to various factors, such as the effects of the torsional modes. The band structure was determined from the measured acceleration and then compared with the analytical band structure. In general, similar observations can be made for the measured band structures for both of the beams. Firstly, the measured band structure was found to be noisy, but a distinct pattern resembling one set of the waves can be found in the Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i

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Fig. 18. Transfer function of polycarbonate and aluminium bi-material beam for error analysis (Beam 1). (a) Point 2, (b) Point 3, (c) Point 4, (d) Point 5, (e) Point 6, and (f) zoomed in portion of Point 2. Analytical band gaps are highlighted in grey.

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Fig. 19. Dispersion curve for Beam 1 for studying the effects of moving accelerometer. (a) Wave A (real part), (b) Wave A (imaginary part), (c) Wave B (real part), and (d) Wave B (imaginary part). Analytical band gaps are highlighted in grey.

measured values. The scatter in the measured band structure was caused by the measurement of one set of waves (the evanescent waves) being sensitive to noise and the effects of the torsional modes being measured by the accelerometer. As the evanescent waves will experience large amounts of attenuation across all frequencies, the band gaps of the beams are determined by the band structure of the propagating waves (Wave A), which was found to agree with the analytical solution. However, several discrepancies can be found around the natural frequencies of the vibration modes that were not considered in the analytical solution. This result highlights a disadvantage of the proposed method, which is the sensitivity to unaccounted modes of vibrations. This problem can be alleviated in future studies by careful placements of the shaker and the use of contactless measurement techniques, such as laser vibrometer. Furthermore, a key shortcoming of the method is that it is unable to reject errors and a future development of the method that can make use of some form of regression analysis will greatly improve the technique. The measured band structure was used to predict the transmissibilities of the final two unit cells (the eighth and ninth unit cells) of Beam 2 in order to validate the method further to determine the transfer function from the measured band structure. The transmissibilities were then calculated by using the measured transmissibilities at the second to fourth unit cells and the band structure measured from the transmissibilities of the second to seventh unit cells. The predicted transmissibilities for the eighth and ninth unit cells were then compared with the measured transmissibilities and the results were found to be in agreement except at the frequencies where the measured result of the band structure and regions was inaccurate. Lastly, a study was performed to determine the effects of the potential sources of errors. Based on the study, the errors were deduced to be caused by a combination of measurement noise, the relocation of the accelerometer with finite mass, and the Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i

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Fig. 20. Dispersion curve for Beam 1 for studying the effects of torsion. (a) Wave A (real part), (b) Wave A (imaginary part), (c) Wave B (real part), and (d) Wave B (imaginary part). Analytical band gaps are highlighted in grey.

torsional modes. Despite the errors in the measurements, the dispersion curves measured for the propagating waves were fairly accurate, allowing for the determination of the band gap. Conversely, the evanescent waves cannot be measured accurately with these sources of error, which suggests that the WSM cannot be used to study the evanescent waves.

Acknowledgements The strong support from The Aviation Industry Corporation of China (AVIC), First Aircraft Institute (FAI) and Beijing Aeronautical Manufacturing Technology Research Institute (BAMTRI) for this funded research is much appreciated. The research was performed at the AVIC Centre for Structural Design and Manufacture at Imperial College London.

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Please cite this article as: L. Junyi, et al., Measuring the band structures of periodic beams using the wave superposition method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.07.005i