Flexural band gaps and response attenuation of periodic piping systems enhanced with localized and distributed resonators

Novel metamaterial concepts can be used to economically reduce flexural vibrations in coupled pipe-rack systems. Here, we model pipe on flexible supports as periodic systems and formulate dispersion relations using Floquet-Bloch theory which is verified by a finite element model. Owing to the flexibility of the coupled system, a narrow pass band is created in low frequency regime, in contrast to the case of pipe without any rack. Two types of vibration reduction mechanisms are investigated for pipe with different supports, i.e. simple and elastic support. In order to tune the band gap behaviour, lateral localized resonators are attached at the centre of each unit cell; conversely, the lateral distributed resonators are realized with a secondary pipe existing in the system. The results reveal that both Bragg and resonance type band gaps coexist in piping systems due to the presence of spatial periodicity and local resonance. Although, the response attenuation of a coupled pipe-rack system with distributed resonators is found to be little lower than the case with the localized one, the relatively low stiffness and damping values lead to cheaper solutions. Therefore, the proposed concept of distributed resonators represents a promising application in piping, power and process industries.

In recent studies, metamaterial beams, shafts and rods endowed with periodic resonators have been investigated. In order to filter undesired longitudinal 17 , flexural [18][19][20][21][22][23][24][25][26] or torsional 27,28 waves, a periodic structure can be equipped with resonator units that entail new band gaps different from those produced by Bragg scattering 7 . The introduction of damping devices in such resonator units contributes to energy dissipation, thereby reducing vibration amplitude 29 . In order to achieve an efficient energy dissipation, the lateral localized resonators (LLRs) or tuned mass dampers (TMDs) have to be optimally designed 30,31 . This is achieved by adjusting their frequency and damping ratios so as to minimize some significant response quantity of the main system, e.g. displacement.
In the context of vibration mitigation in pipes, tuned mass dampers with different energy dissipation mechanisms such as material damping in the damper element of TMD 32,33 , fluid damping in TMD mass 34 or impact of TMD mass with a dissipative surface 35 were developed. All adopted damper technologies require the use of an external mass, indeed, which needs to be connected to a pipe using a spring-damper system. Clearly, this solution results to be less practical.
In a typical pipe-rack system, it is common to have pipes of different dimensions. It may thus be possible to conceive a pipe of smaller cross section, see P 1,α in Fig. 1b, as a lateral distributed resonator (LDR) system. The dynamic characteristics of P 1,α and its effects on a coupled pipe-rack system are different from those of LLRs and are the topics explored in the paper further in conjunction with the periodicity effects.
Along these lines, the propagation of flexural waves is studied in a pipe-rack system similar to the case study shown in Fig. 1. Two models are considered in this regard: (i) a pipe without rack, i.e. a pipe on simple supports -Type #1-; and (ii) a pipe with a rack, i.e. a pipe on elastic supports -Type #2-. Type #2 support approximates a realistic scenario while Type #1 represents an extreme case. As a result, flexural wave propagation in an undamped long elastic pipe P on Type #1 and Type #2 supports is examined, both analytically and numerically. Thus, in order to determine analytical dispersion relations, the Floquet-Bloch theory of periodic systems is employed.
Nonetheless, it is not straightforward to derive similar dispersion relations when material damping in a pipe P is taken into account or when either a LLR or a LDR (P 1,α ) is attached to P. Hence, a finite element (FE) model is set for these studies, the accuracy of which is verified by comparing an undamped finite element model of P with analytical results. Optimal stiffness and damping values are then designed for the case when P is connected to a LLR and to LDR or P 1,α . Moreover, the effect of damping in resonators on band gap characteristics is investigated by means of the transfer function of P provided by the FE model of the coupled system. Finally, the vibration attenuation of P by means of LLRs is compared with the case when LDRs with the same mass ratio α, i.e. P 1,α is used. Though the performance of a coupled pipe-rack system with LDRs is less efficient than the case with LLRs, relevant low stiffness and damping values lead to cost-saving solutions.

Results
Wave propagation in an undamped pipe. The dispersion relation for an undamped pipe P of unit cell length l, cross sectional area A, moment of inertia I, density ρ and Young's modulus E with Type #2 support condition shown in Fig. 2a is derived based on the Floquet-Bloch theorem 36,37 . The analytical expression of dispersion relation between the circular frequency ω and the propagation constant μ (i.e., ikl) is given by  and See the section 'Methods' for detailed derivation. The analytical dispersion relation given in Eq. (1) is obtained by solving Eq. (19).
The behaviour of the pipe-rack coupled system can be characterised by these dispersion relations. Equation (1) is quadratic in coshμ and thus yields two pairs of distinct roots of μ; ±μ 1 and ±μ 2 for each frequency ω. Positive and negative signs of μ describe the same characteristics of wave motion travelling in opposite directions. Generally, μ is complex and can be written as where, the real part describes rate of attenuation of amplitude and imaginary part γ imparts information about the phase difference of travelling wave between two adjacent unit cells. If the Bloch wave number of a freely propagating non-decaying wave (δ = 0) is k, then www.nature.com/scientificreports www.nature.com/scientificreports/ and is related to the corresponding wavelength λ as Based on the nature of μ, there are three types of wave. For δ > 0, there will be a decay in the amplitude and hence no energy flows in the direction of wave propagation, and are defined as attenuating/evanescent waves. In this case, the adjacent unit cells vibrate either in phase or out of phase. On the other hand, if μ is purely imaginary, energy flows in the direction of propagation and the waves pass without any attenuation thereby exhibiting only pass band in the dispersion curves. For a complex μ, a part of the energy propagates while the remaining gets attenuated, which results in the occurrence of both pass and stop bands in the dispersion curves.
When → ∞ K v and = m 0, the analytical expression of dispersion relation given in Eq. (1) leads to the case of P on Type #1 support. The resulting expression of dispersion relation is given by, which is similar to those found in literature 3 .
For the analytical flexural wave propagation study, a pipe P with = l m 6 , outer diameter of . mm 406 40 and thickness . mm 7 92 is used. Young's modulus and density are assumed to be 200 GPa and 7800 kg/m 3 , respectively.
Equation (5) is used to obtain the variation of k and μ with frequency of the wave f = 2π/ω, and is shown in Fig. 3a,b, respectively. Frequency ranges of first two stop bands are − . Hz [0 31 36] and . − . Hz [69 63 131 2] and are represented by yellow shaded region in Fig. 3a. The pass bands, between these frequencies, are represented by grey shaded region in Fig. 3b,c. Now, wave propagation characteristics in pipe P with Type #2 support as shown in Fig. 2a is examined. For this case, a typical concrete rack structure of C50/40 class is considered. Each frame of the rack is 6.5 m wide and 7.3 m tall with 6 m spacing between adjacent frames along the length of P. Each frame consists of two storeys, placed at a level of 7.3 m and 5.3 m from the ground. Each column of the rack is of 600 mm 2 . The pipe-rack structure consists of 40 frames of which the first 10 are shown in Fig. 1b. A simplified numerical model using spring K v and lumped mass m, as shown in Fig. 2a, is made such that its first mode matches with the first lateral mode of the pipe-rack structure obtained using a complete FE model. It is observed that the first mode occurs at 4.45 Hz and based on the corresponding mass participation, m and K v are calculated to be 22.88 T and 17.9 MN/m, respectively.
Based on the dispersion relation from Eq. (1), the variation of k and μ with frequency of the wave f is obtained and is shown in Fig. 4a,b, respectively. In contrast to Type #1 support, a narrow pass band is generated near 4.45 Hz, which is the first predominant natural frequency of the rack structure. In the frequency range from 0 to  Vibration in a controlled periodic pipe. An identical single mass LLR of mass α m t , , stiffness α k t , and damping coefficient α c t , is attached as a substructure at midpoint (l/2) of each unit cell of P for both Type #1 and Type #2 supports. Similarly, α P 1, of mass ratio α which is present in the rack is attached to P using spring-damper system with stiffness α k P 1, and damping coefficient α c P 1, at the centre of each unit cell. Figure 2b shows the configuration of P with LLRs for Type #2 support while Fig. 2c shows the case when α P 1, is attached as LDR. Material damping ratio ξ for both the pipes P and α P 1, is assumed to be 0.02. In order to assess the performance, three mass ratios of 0.05, 0.16 and 0.25 are chosen and the corresponding dimensions of α P 1, are detailed in Table 1. Optimization is performed to reduce the vibration response till Hz 60 using Eq. (21), based on which the optimal α k t , and α c t , for Type #1 and Type #2 supports are calculated. Equation (22) is used to calculate the corresponding performance index Z for all the cases, a smaller value of which denotes better performance in terms of vibration reduction. Similarly, the optimal α k P 1, and α c P 1, are evaluated for the case when α P 1, is connected to P. Table 2 contains the optimal values for both scenarios. The equivalent frequency α f t , of the LLR and the corresponding damping ratio ξ α t , are provided in Table 3. Since α P 1, is a continuous system with multiple modes, its frequency and damping ratios are not reported.
The frequency response of only the case with α = . 0 16 is reported here as others show similar behaviour. In order to understand the effect of damping on band gaps when a LLR is used, three cases are considered for both types of supports; (i) material damping in P ξ = for Type #2). Figure 5a shows the response for above three cases with Type #1 support while Fig. 5b shows the same for Type #2 support.
Hz [30 72 42 42] is the resonance stop band. The reduction in amplitude of vibration of P is compared when α= . P 1, 0 16 is used as LDRs instead of the LLR of α = .
0 16. For this, the corresponding frequency response plots are shown in Fig. 6a,b, for Type #1 and Type #2 supports, respectively. As expected, significant reduction in vibration is achieved for the case of LLR, while α P 1, is found to be less efficient for the same mass ratio. More precisely, with LLRs/LDRs attached to the pipe P, the total mass of the system is increased and natural frequencies of the system are decreased. As a consequence, the width of a stop band is reduced and this reduction is higher in case of LLRs as compared to the LDRs. In fact, in the case of LLRs the mass is concentrated in the middle of each pipe span and the relevant participating mass is higher than the case with a distributed mass. Furthermore, both LLRs and LDRs are introduced to reduce the vibration amplitude of propagation waves in the first pass band for Type #1 support and in the second pass band for Type #2 support, respectively. This is achieved by means of the optimal stiffness and damping values reported in Table 2 for α = .
0 16. Owing to dynamic effects of LLRs and LDRs, one can note the shrinking of stop bands and reduction of amplitudes in pass bands in Fig. 6.

Discussion
In order to reduce flexural vibrations in coupled pipe-rack systems of LNG plants, novel and economic ways based on metamaterial concepts are proposed. As a result, flexural wave propagation in periodic piping system is analysed by solving relevant fourth-order equation of motion. Subsequently, the Floquet-Bloch theorem is applied to obtain the analytical dispersion relation, which was used to validate the results provided by a numerical model. The propagation characteristics of pipe P for both types of support conditions, simply supported -Type #1-and with elastic supports -Type #2-are identical except that in case of Type #2 supports, where a new narrow pass  Pipes without lateral localized resonators (LLRs) at the centre of each unit cell exhibit only Bragg band gaps. Conversely, the adoption of LLRs without damping entails a new band gap located near the natural frequency of LLRs which can be observed in both Fig. 5a,b for Type #1 and Type #2 support, respectively. The results reveal that both Bragg and resonance type band gaps coexist in piping systems due to the presence of spatial periodicity and local resonance. In addition, the introduction of LLRs increases the whole mass of the system, which results in decreasing the main natural frequencies of the coupled pipe-rack system. More precisely, in the case of Type #2 supports the first Bragg stop band remains the same, while the second stop band shifts to the low frequency range as depicted in Fig. 5b. The presence of damping in LLRs smoothens and lowers the transmission ϕ T of P and widens the band gap. High damping in LLRs causes the band gap to vanish 21,22 , but can significantly reduce vibration amplitude.
Further, a new way of vibration suppression was investigated by the attachment of a smaller cross section pipe, the so called lateral distributed resonators (LDRs), in parallel to the main pipe P. It was observed that, with an increase of the mass ratio α between resonators and pipe, both LLRs and LDRs ( α P 1, ) performed better. This is evident from Table 2, which shows a decrease of the performance index Z, defined in Eq. (22), with an increase of α. Also, the optimal values of α k t , and α c t , for LLRs increase with an increase of α. For the same α, it was found that Z corresponding to the case LDRs was greater than that of the case with LLRs; this can be inferred from both Table 2 and Fig. 6. In fact, for any value of α, the optimal α k P 1, and α c P 1, for α P 1, is less than that for the LLRs case. This is because α P 1, is a continuous system and, therefore, the whole mass cannot be mobilized for any frequency: this leads to a lower effective mass ratio. Conversely, for the corresponding LLRs case, the complete mass contributes to the frequency for which it is designed 38 . Clearly, the trend of ϕ T shown in Fig. 6 for α= . P 1, 0 16 cannot be utilized for other pipes endowed with α = .
0 16. This is because the dynamic characteristics of α= . P 1, 0 16 depends on radius/thickness ratio. Thus, to determine optimal parameters for other pipes, a rigorous optimization has to be performed again.
In sum, even though the performance of a coupled pipe-rack system with LDRs is less efficient than the case with LLRs, the relevant low stiffness and damping values lead to cheaper solutions. As a result, the adoption of pipes α P 1, represents a promising application. Eventually, the enhancement of these results by means of nonlinear devices/mechanisms deserves further studies.

Methods
Derivation of dispersion relations. The system shown in Fig. 7a is considered for flexural wave analysis which consists of two unit cells each of length l. The undamped pipe P is assumed to be an Euler-Bernoulli beam, the equation of which is given as, www.nature.com/scientificreports www.nature.com/scientificreports/ where ρ and E are the density and modulus of elasticity of the material, respectively. A is area of cross-section and I is the second moment of inertia of the beam. w x t ( , ) is the transverse displacement, and x represents the spatial coordinate along the length of beam. The substitution of steady state solution = ω w x t w x e ( , ) ( ) i t in Eq. (6) leads to, where ω is the circular frequency. The solution of Eq. (7) can be written as, Applying Floquet-Bloch theory on each unit cell as shown in Fig. 7b, the transverse displacements of nodes + i 1 and − i 1 are related to those at node i as where, k represents one dimensional Bloch wave number, l is the distance between supports and i is −1 . The term ikl in Eq. (9) is called propagation constant μ. The similar relations are used for rotations. The constants A 1 , A 2 , A 3 and A 4 in Eq. (8) are computed by applying boundary conditions shown in Fig. 7d, and are then used to determine the shear forces V and bending moments M on either side of node i. Figure 7c shows the sign convections for shear forces and bending moments. The expression for dynamic compliance coefficients 39  The kinematic compatibility condition for rotation is given by i i and for equilibrium at node i (Fig. 7e), the bending moments and forces equations are given as v i 2 from the above Eqs. (12), (13) and (14), the linear homogeneous equations are obtained in terms of w i and ϕ + i as follows, where, K v and m are lateral stiffness of rack column and lumped mass of the rack, respectively. Equations (15) and (16) can be written as, = Hu 0 (17) where, ϕ = + u w ( , ) c t , . The values of these bounds are chosen such that the optimal α k t , and α c t , do not take unrealistic values and thus result in a faster optimization. A similar genetic algorithm-based optimization is used to determine the optimal spring-damper parameters ( α k P 1, and α c P 1, ) when α is used instead of a LLR as shown in Fig. 2c.