Thin-walled cylindrical shells are widely used in industries, for example, the main parts of aircrafts, rockets, and submarines are all made of cylindrical shells. The radiation noise generated by the vibrating cylindrical shells may harm precise equipment inside the aircrafts, and even cause fatigue failure of structures [1]. As a result, cylindrical shells must have good vibration and acoustic performances while meeting static bearing capacity. To fulfill such demands, many methods have been proposed to reduce the vibration and the corresponding radiation noise of cylindrical shells. Boily et al [2] investigated the impact of viscoelastic and porous materials on vibration and noise reduction of cylindrical shells. Ji et al [3] proposed a method to control the vibration of pipelines by using magneto-rheological (MR) vibration reduction technology. Zhang [4] proposed to control the vibration of the cylindrical shell by periodically or non-periodically–attached dynamic vibration absorber (DVA). In addition to damping and absorbing vibration, there are also vibration control methods. Wang et al [5] investigated the vibration isolation effects of a blocking mass on a double-layer cylindrical shell and demonstrated that it can effectively prevent the transmission of vibration waves. Huang et al [6] used a numerical model to investigate the vibration transmission from vibrating machinery to the cylindrical shell structure through the active vibration isolators. In order to improve the performances and effects of viscoelastic damping layers, Baz et al [7] proposed the concept of an actively constrained damping layer. Ray et al [8] arranged the actively constrained damping layers on a large-scale cylindrical shell, and optimized the parameters and positions of the damping layer to realize good vibration reduction effects. However, the aforementioned methods and many others only have good effects at middle or high frequencies; controlling low-frequency vibration of cylinder shells is still a huge challenge.

During the last decades, the emerged elastic metamaterials provide possibility to dealing with the low-frequency vibration. Such manmade materials show bandgaps at low frequencies generated by local resonators. The elastic metamaterial with local resonators is first proposed by Liu et al [9] in 2000. After that, researchers studied the unprecedented properties of metamaterials, such as negative mass density [10–12], negative modulus [13, 14], and even simultaneous negative density and modulus [15, 16]. From the perspective of materials, bandgaps are the consequences of negative effective parameters. Within these bandgaps, propagation of elastic waves is prohibited. Consequently, the vibration of structures made by metamaterials is significantly reduced. Chen et al [17] proposed the addition of local resonators into a sandwich beam and studied the bandgap effects in this metamaterial beam, results show the waves are significantly attenuated. Claeys et al [18] designed a local resonant metamaterial plate and investigated the noise radiation properties within the bandgaps, good vibroacoustic behaviors of this plate are observed. Jung et al [19] explored the use of metamaterials to reduce the low-frequency noise radiation of an automobile dash panel structure, numerical and experimental results verified the idea. Recently, metamaterials are also used in shell structures. Droz et al [20] mounted local resonators on a curved panel to enhance the sound transmission loss at the ring frequency. Jin et al [21] designed a metamaterial cylindrical shell by introducing local resonators into unit cells of a traditional cylindrical honeycomb sandwich structure, and they studied the bandgap phenomena in such metastructures and found that within the bandgaps, vibration of the cylinder shell is considerably mitigated. It should be noted that metamaterials usually suffer the drawback of narrow sizes. To broaden the bandgaps, metamaterials with multiple resonators in a single unit cell [22–26] or gradually varying in space [27, 28] were proposed and studied. The working frequency regions of these metamaterials are enlarged to some extent, however, at the cost of adding considerable extra mass. In summary, even though metamaterials with local resonators are promising candidates for reducing low-frequency vibration, they either have narrow working frequencies or would add too much mass into the original systems, which makes them unsuitable to control multi-modal resonances of thin-walled cylindrical shells at low frequencies.

In addition to the previously introduced metamaterials, which are composed of passive mechanical local resonators, piezoelectric metamaterials based on resonant shunts are also widely studied. These so-called local resonant piezoelectric metamaterials (LRPMs) generate bandgaps via piezoelectric materials and electronic resonators. Throp et al [29] first proposed to periodically distribute piezoelectric patches with resonant shunts on a rod and observed bandgaps induced by shunts. Spadoni et al [30] extended this concept to two-dimensional structures and developed piezoelectric metamaterial plates. Csadei et al [31] designed a method to predict the attenuation properties of the periodically assembled unit cells, and verified it experimentally. Furthermore, Airoldi and Ruzzene [32] studied a beam with periodically shunted piezoelectric patches, and they found that the equivalent stiffness of the metamaterial beam is significantly affected by circuit parameters and has resonance characteristics near the circuit resonant frequency, leading to bandgaps. Therefore, from the perspective of metamaterials, periodical piezostructures with resonant shunts are new types of metamaterials. To broaden the overall bandgap size, Sugino et al [33] proposed a transfer function to realize multiple resonances in a piezoelectric metamaterial beam. Wang et al [34] designed digital circuits using separated patches as sensors and actuator to realize multiple resonances in a piezoelectric metamaterial beam. However, these multi-resonant metamaterials are limited to beam structures. Recently, Yi et al [35] developed a general method to design multi-resonant piezoelectric metamaterials based on digital circuits, multiple bandgaps and the wave isolation effects are demonstrated using a plate structure. Theoretically, the method developed in [35] can be expanded to design any type of multi-resonant metamaterial structures, including cylindrical shells, which gives a new possibility to control broadband low-frequency vibration of such complex structures.

In this study, a novel cylindrical meta-shell is designed based on piezoelectric materials and digital circuits. A transfer function to realize multi-resonance developed by the authors in [35] is implemented in the digital circuit. A method was developed to optimizing the transfer function for realizing the best vibration reduction effects at low frequencies. The vibration characteristics of such metashells are numerically studied, and multi-modal vibration reduction effects are clearly demonstrated. The rest of this study is organized as follows: Section 2 introduces the conception and major components of the meta-shell with piezoelectric patches and digital circuits; in Section 3, a reduced and corrected model is developed to efficiently calculate the forced response of piezoelectric metashells; an algorithm to optimize the parameters in the transfer function is proposed in Section 4; Section 5 verifies the excellent multi-modal vibration reduction performances at low frequencies; finally, Section 6 presents conclusions of this work.

In this section, the model of the studied cylindrical meta-shell with piezoelectric materials and digital circuits is presented. Such a meta-shell is composed of four main parts: (I) a host passive shell structure, (II) piezoelectric patches distributed on the outer surface of the host shell, (III) digital circuits connecting to the patches, and (IV) a suitable transfer function implemented in the digital circuit.

This section develops a reduced model to efficiently calculate the vibration responses of the cylindrical piezoelectric meta-shell. The reduced model is obtained in three steps. First, the piezoelectric meta-shell is modeled using the finite element method. Second, the whole model is reduced using the modal synthesis method. Last, the reduced model is corrected through the modification of the intrinsic capacitances of the patches.

The equilibrium equations for the discretized fully coupled piezoelectric system are [36]: M d d d ¨ + K d d d + Θ d v V = F − Θ d v T d + C V = − Q . (4) Here, M d d and K d d represent the mass matrix and stiffness matrix in the short-circuit state, respectively, Θ d v and C are the electromechanical coupling coefficient matrix and the blocked intrinsic capacitance matrix, respectively, d and V represent the structural and voltage DOFs, respectively, and F and Q are the mechanical forces and charges, respectively. The Kirchhoff’s current law must be obeyed at the joints where circuits are coupled with the piezoelectric patches, as shown in Eq. 4.

The reduced model is obtained through a transformation between the displacement d and a set of modal coordinates q . d = Φ q . (5) Here, Φ = [ ϕ 1 , ϕ 2 , … ϕ m ] . ϕ i is the ith natural mode of the piezoelectric system under the short-circuit condition with specific homogeneous Dirichlet boundaries, and it is obtained by solving the following eigenvalue problem: ( − ω i 2 M d d + K d d ) ϕ i = 0 . (6) Here, ω i is the corresponding natural frequency. The modes are mass-normalized, resulting in: Φ T M d d Φ = E , (7) Φ T K d d Φ = Λ = d i a g ( ω i 2 ) . (8) Here, E is the identity matrix. Using Eqs 5, 7, 8, the governing Eq. 4 is represented in modal coordinates as: q • • + Λ q + Φ T Θ d v V = Φ T F − Θ d v T Φ q + C V = − Q . (9)

Only a few modes in modal matrix Φ will be retained, and the number is much smaller than that of the system’s DOFs. Thus, the number of equations in (9) is largely reduced.

However, the reduced model in Eq. 9 cannot adequately capture the system’s piezoelectric behaviors because the truncation of the higher-order modes will cause a static reduction error [36]. The static error will result in a non-negligible electrostatic voltage error when compared to the full model, it is as follows: T e r r e = T f e − T r e = Θ d v T ( K d d − 1 − Φ Λ − 1 Φ T ) Θ d v (10) Here, T f e and T r e are the electrostatic transfer matrices between V and Q of the full and reduced models, respectively.

The reduced model is corrected by modifying the blocked intrinsic capacitance matrix C to produce more accurate voltage responses. The voltage responses are corrected by ensuring that the voltage outputs V( i ) of the ith piezoelectric patch in the corrected reduced model and the full model are consistent when the same static electric input Q( i ) is applied to this patch. According to Eqs 4, 9, such requirements are fulfilled by modifying the capacitance matrix C ∗ in the corrected model as: C ∗ = d i a g ( Θ d v T ( K d d − 1 − Φ Λ − 1 Φ T ) Θ d v ) + C . (11) C ∗ is still a diagonal matrix, the corrected reduced model is obtained by replacing the C in Eq. 9 with C ∗ . q • • + Λ q + Φ T Θ d v V = Φ T F − Θ d v T Φ q + C ∗ V = − Q . (12)

The reduced model obtains an excellent accuracy after error correction and improves computational efficiency, and it provides an efficient tool to calculate the dynamic responses of the piezoelectric meta-shell, which is a very important step in the optimization algorithm developed in Section 4.

For the transfer function in Eq. 1, ω p , i and β i affect the frequency and strength of circuit’s resonance, respectively [35]. Therefore, to realize optimal vibration reduction effects near a target resonant mode, the values of ω p , i and β i need to be optimized, a method is developed in this section for achieving such purposes.

As analyzed in [35], if a transfer function contains n poles and these poles are well separated, they have negligible influences on each other, which means that the values of ω p , i and β i for controlling the ith mode can be optimized individually. On the other hand, to minimize the time consumption, during the optimization process, the maximum value of the dynamic response of a carefully selected point within a frequency band near the resonance frequency of the target mode is used as an index to find the optimum design parameters. It should be noted that the selected point cannot be on the nodal lines of the target mode’s modal shape.

When the target modes are selected, we can determine the number of poles n needed in the transfer function. The values of ω p , i and β i for the ith pole can be optimized through several steps, as illustrated in Figure 2. First, determine the frequency range [ f i , L ,   f i , R ] . The selection of the frequency range [ f i , L ,   f i , R ] is tricky, if it is too wide, the simulation time will be long, on the opposite, if it is too narrow, the resonance frequency of the target mode may be out of this range, and the algorithm will fail to find the optimum values. Therefore, a tradeoff between the efficiency and accuracy must be made. In this work, f i , L and f i , R are set as the left and right anti-resonance frequencies near the target mode, respectively, an example is given in Figure 3. Second, determine the searching ranges for ω p , i and β i . ω p . i is the resonance frequency of the digital circuit, so it should be within the simulation frequency band, therefore, it is searched within [ 2 π f i , L ,   2 π f i , R ] . β i influences the resonance strength of the digital circuit, and it should be a small value, consequently, the searching range of it is given as [ 0 ,   1 ] . Third, implement a n-pole transfer function in the digital circuit, let the parameters of the poles except for the ith one be zero, sweep the parameters ω p , i and β i within the determined ranges, simulate the frequency response function H ( f ) at a selected point on the shell using the reduced model in Eq. 12, and calculate the maximum value max ⁡ ⁡ ( H ( f ) ) . Fourth, choose the values of ω p , i and β i corresponding to the minimum max ( H ( f ) ) as the optimum values. Repeat the aforementioned procedures for n times, and all the optimal parameters can be found.

In this section, the multi-modal vibration reduction effects of piezoelectric cylindrical meta-shell are numerically verified. Figure 4 shows the finite element model of the studied piezoelectric meta-shell. There are 8∗5 = 40 piezoelectric patches glued on the surface of the passive shell. The geometry and material parameters of the cylindrical piezoelectric meta-shell are the same as those in Table 1. The convergence of the model is verified; refining the meshes to double the number of DOFs only changes the calculated natural frequencies of the first 10 modes by less than 0.2%. A harmonic load F is applied at point O to mimic a source of disturbance.

In this study, a cylindrical meta-shell with piezoelectric materials and digital circuits is designed, and the multi-modal vibration reduction effects of it are demonstrated. The mechanical parts of the meta-shell include a host cylindrical shell, piezoelectric patches distributed on the outer surface of the shell, and digital circuits shunted to the patches. A suitable transfer function is implemented in the digital circuit to generate multiple resonances. To analyze the dynamic responses of the designed piezoelectric cylindrical meta-shell, a reduced model is obtained based on the modal truncation technique and the accuracy of it is corrected by modifying intrinsic capacitances of the patches. An optimization algorithm is developed to design the optimum parameters in the transfer function for the purpose of vibration reduction. Finally, cases are studied to verify the multi-modal vibration reduction abilities of the meta-shell. The results clearly demonstrate that by designing the parameters of the transfer function, multi-modal vibration reduction effects can be achieved for desired resonant modes, the peak magnitudes of these modes are suppressed by more than 30 dB. Therefore, this novel piezoelectric cylindrical meta-shell could open new opportunities in vibration mitigation of transport vehicles and underwater equipment.