With the development of naval architecture and ocean engineering with high-speed, large-scale and large-tonnage, vibrations in ship and offshore structures owing to various ocean environmental loads and excitations of power systems become increasingly serious, can reduce productivity, endanger safety, affect service life of the structure and even discomfort of crews (Hirdaris et al., 2014; Murawski and Charchalis, 2014; Kandasamy et al., 2016). Therefore, it is necessary to eliminate or reduce vibration and noise of ship and offshore structures and develop low-noise structural design technology.

Many scholars have done a large number of studies to control vibration and noise of ship and offshore structures (Wang et al., 2013; Lan et al., 2014; Wang et al., 2016; Gripp and Rade, 2018). From their studies, there are three main methods to control vibration and noise. The first way is to control the source, design and manufacture power systems with low noise, low vibration or even silence; the second is to control the propagation path through isolation, absorption, and attenuation methods; the third is to protect the receiver (Wu et al., 2014; Liu et al., 2015; Shi et al., 2016; Som and Das, 2018). As the development of naval architecture and ocean engineering with high-speed, large-scale and large-tonnage, it has become increasingly difficult to reduce noise from the source. Furthermore, the techniques to protect the receivers are unreasonable as the receivers are often moving as well as the rapid developments of the underwater acoustic detection technologies. Therefore, trying to reduce the vibration and noise in the propagation path is a more sensible choice (Ibrahim, 2008; Toky et al., 2020; Yaman et al., 2021). The mainly ways to control vibration and noise can be divided into passive control schemes, semi-active control schemes, and active control schemes (Keir et al., 2005; Ou et al., 2007; Liu et al., 2017; Enferadi et al., 2019; Han et al., 2019; Li and Yang, 2020). Although the existing traditional vibration reduction techniques can control the vibration and noise of ships and offshore structures to a certain extent, it also has some limitations, including the equipment mass must be larger and the narrow frequency band in passive control, the damping vibration attenuation only have an impact on the medium and high frequencies, as well as the system complexity and narrow band characteristics of active control. In addition, it is necessary to strengthen the research on low frequency vibration and noise control of ships and offshore structures, as the vibration and noise in the low frequency range has strong penetration and slow dissipation during propagation (Waye, 2011; Kim et al., 2019). Beam is one of the most important basic structures in engineering and are widely used in civil engineering, aerospace as well as naval architecture and ocean engineering. And stiffened plates, especially the bi-directionally orthogonal stiffened plates are widely used in deck, bottom and side structures of ships and offshore structures. The stiffened plates can be simplified to a multi-span beam model with periodic elastic supports and then into an elastic foundation beam model to study their static and dynamic characteristics (Chen and Chen, 1991). In addition, many slender stiffened plate structures in ships and offshore structures can be simplified into beam models for low frequency flexural wave vibration analysis. Thus, studying the dynamic characteristics of beams is of great significance to the dynamic performance and low noise structural design of ships and marine structures. Many scholars have done a lot of studies to investigate the wave band gaps and propagation characteristics in phononic crystals, acoustic metamaterials and other periodic structures in recent decades (He et al., 2017; Lee et al., 2018; Nateghi et al., 2019). Periodic structures have been introduced into the structural vibration reduction design due to the advantage of band gap characteristics.

Periodic structures have band gaps properties, which the propagation of waves in the periodic structures is forbidden in specific frequency ranges. Due to the various unique physical properties, periodic structures are widely used in various engineering practices such as structural strength enhancement, vibration absorption and vibration control, thus, a large number of studies have been carried out on the band gaps and wave propagation and attenuation characteristics of various periodic structures (Dupont et al., 2019; Muhammad and Lim, 2019; Xu et al., 2021). The existing periodic structure studies reveal two kinds of band gap formation theories: local resonance band gap and Bragg scattering band gap. Mead and his team members have conducted a lot of researches on the wave propagation characteristics in periodic beams since the 1960s (Mead, 1970; Mead, 1996). Kushwaha et al. inferred phononic crystals from the studies of photonic crystals and discovered the band gap phenomenon of phononic crystals (Kushwaha et al., 1993). Liu et al. fabricated a local resonant periodic structure by the idea of localized resonant structures, and the theoretical and experimental studies have verified there is a low-frequency local resonance band gap in this type periodic structure which is much lower than the traditional Bragg gap frequency (Liu et al., 2000). It provides a theoretical foundation for the application of low-frequency band gap characteristics of macroscopic periodic structures.

In the past two decades, the elastic wave band gap characteristics and propagation and attenuation characteristics of various periodic structures have been investigated. Guo and Fang (2014) and Li and Guo (2016) analyzed the longitudinal wave dispersion relations and band gap characteristics in periodic quaternary rods and rod-type piezoelectric periodic structures, respectively. The research results promoted the design of rod-type periodic structures for vibration isolation/control applications. Richards and Pines (2003) utilized the property of a periodic drive shaft that generated stop band gap and pass band regions in the frequency spectra, both the analytical and experimental results indicated that the proposed periodic drive shaft can effectively reduce the transmitted vibration generated by gear mesh contact dynamics. Shen et al. (2012) proposed a periodic shell made of functionally graded material and then investigated the mechanism of wave propagation and vibration transmission in the shell by illustrating the pass/stop band frequency ranges of the periodic shell. An et al. (2018) numerically studied the band gap characteristics of radial wave propagating radically from the inner circle of a two-dimensional cylindrical shell structure with radial and circumferential periodicities. Results showed that radial wave has significant attenuation in band gap frequency region. Sharma and Sun (2016) investigated the low frequency wave propagation behaviours of sandwich beams containing periodically embedded internal resonators, and they demonstrated that local resonance and Bragg band gaps coexist in the proposed periodic sandwich beams. Liu and Yang (2017) analyzed the characteristics of acoustic wave transmitting in a metamaterial seawater pipe which consists of a uniform pipe with air-water chamber Helmholtz resonators mounted periodically along its axial direction, the results showed that the proposed metamaterial seawater pipe could generate a wide band gap in the low-frequency range and rendering the propagation of the frequency range acoustic waves in the piping system dampened spatially. Faiz et al. (2020) theoretically and experimentally investigated the waveguiding and filtering properties of a two-dimensional phononic crystal slab. They verified that the proposed device shows a complete band gap and the Lamb wave in the phononic crystal slab can be suppressed significantly. Zhou et al. (2014) studied the flexural wave band gap and attenuation characteristics in a periodic stiffened-thin-plate with the help of center-finite-difference-method. Results showed that the proposed periodic structure can yield complete band gaps and the wave propagation is forbidden. Zuo et al. (2018) investigated the flexural wave propagation and band gaps of a locally resonant phononic crystal to achieve a wider band gap and a lower cut-on frequency. This research provided an approach for the application of locally resonant phononic crystal in vibration and noise control.

As a preliminary study of periodic structure band gaps characteristics and low-noise structural design of ship and offshore structures, a simple physical model is proposed in this paper for analyzing the dynamic characteristics of periodic support beams and low-frequency flexural wave vibration of slender stiffened plate structures in ship and offshore structures. The proposed physical model is an infinite long homogeneous beam coupled with periodic oscillators which is a periodic beam-oscillators coupling system composed of homogeneous beam and periodic local oscillators. Based on the Timoshenko beam theory and its wave form solution, the calculation method for the flexural wave band gaps in the proposed infinite long homogeneous beam coupled with periodic oscillators is established by the method of reverberation-ray matrix combined with Bloch theorem. The method of reverberation-ray matrix is a semi-analytical method and the advantages of the present method lie in its simplicity, clarity, and accuracy (Guo and Fang, 2011). The rest of this article is organized as follows: First of all, the calculation formulation for analyzing the flexural wave band gaps of the homogeneous beam coupled with periodic oscillators is established by using the method of reverberation-ray matrix, as detailed in Section 2. The theoretical calculation results and discussions of the band gap characteristics are given in Section 3, which the flexural wave band gaps calculation method is verified by structural flexural wave vibration transmission characteristics based on finite element method, as shown in Section 3.2. And then the influences of various structural and material parameters of the homogeneous beam coupled with periodic oscillators on the band gap characteristics are respectively investigated in Section 3.3. Conclusions are given in Section 4.

The proposed simple physical model considered in this paper is constituted of an infinitely long homogeneous beam coupled with periodic local oscillators in parallel connection as illustrated in Figure 1A , which is a periodic beam-oscillators coupling system. The area surrounded by a red dashed line in Figure 1A is the unit cell of the proposed infinite long homogeneous beam coupled with periodic oscillators and the schematic diagram is shown in Figure 1B, and the periodic beam-oscillators coupling system in a unit cell is drawn in Figure 1C. By periodically repeating and combining the unit cell along the length direction, the proposed infinite long homogeneous beam coupled with periodic oscillators can be obtained. The structural parameters of the proposed infinite long homogeneous beam coupled with periodic oscillators are listed as follows: the spring stiffness k _s , the mass m _s and the damping c _s of the periodic oscillators, lattice constant a, section moment of inertia I, section area A as well as the support stiffness of the elastic foundation k*, and the material parameters of the homogeneous beam including Young’s modulus E, mass density ρ and Poisson’s ratio ν. In advance of investigating the flexural wave band gaps and vibration attenuation characteristics in the proposed infinite long homogeneous beam coupled with periodic oscillators, the governing wave equations should be introduced. The main equations of the flexural wave band gaps calculation method for the proposed infinite long homogeneous beam coupled with periodic oscillators are as follows.

Considering the effects of shear deformation and moment of inertia, the flexural wave vibration governing equations of a homogeneous Timoshenko beam with elastic foundation support can be written as (Li et al., 2014): E I ∂ 2 φ ∂ x 2 + κ G A ( ∂ w ∂ x − φ ) − ρ I ∂ 2 φ ∂ t 2 = 0 (1) κ G A ( ∂ 2 w ∂ x 2 − ∂ φ ∂ x ) − ρ A ∂ 2 w ∂ t 2 + k ∗ w = 0 (2) where G represents the shear elastic modulus, κ denotes the shear correction coefficient, w (x, t) and φ(x, t) are the vertical displacement and rotation angle of the homogeneous beam, respectively.

By eliminating ∂ φ / ∂ x after substituting Eq. 1 into Eq. 2, the flexural wave vibration governing equation expressed only by vertical displacement w (x, t) can be obtained as ∂ 4 w ∂ x 4 + k ∗ κ G A ∂ 2 w ∂ x 2 − ( ρ κ G + ρ E ) ∂ 4 w ∂ x 2 ∂ t 2 + ( ρ A E I − ρ k ∗ E κ G A ) ∂ 2 w ∂ t 2 + ρ 2 E κ G ∂ 4 w ∂ t 4 − k ∗ E I w = 0 (3)

The wave form solution for the vertical displacement in the homogeneous beam can be expressed as w = W ₀ e ^i(kx−ωt). Thus, after omitting the simple harmonic time factor e ^−iωt , the solution of Eq. 3 in the frequency domain can be expressed as follows: w = a 1 e i k 1 x + d 1 e − i k 1 x + a 2 e i k 2 x + d 2 e − i k 2 x (4) where k 1 = − α / 2 + ( α / 2 ) 2 − β and k 2 = − α / 2 − ( α / 2 ) 2 − β are wavenumbers, and where α = − k ∗ κ G A − ω 2 ( ρ κ G + ρ E )   , β = − ω 2 ( ρ A E I − ρ k ∗ E κ G A ) + ρ 2 E κ G ω 4 − k ∗ E I (5) From this, the rotation angle of the homogeneous beam can be written as φ = g 1 a 1 e i k 1 x − g 1 d 1 e − i k 1 x + g 2 a 2 e i k 2 x − g 2 d 2 e − i k 2 x (6) where g _j (j = 1, 2) can be expressed as g j = i k j κ G A E I k j 2 + κ G A − ρ I ω 2 (7)

According to the relationships between the bending moment with the vertical displacement, M = E I ∂ 2 w / ∂ x 2 , and the shear force with the vertical displacement and rotation angle in Timoshenko beam, V = κ G A ( φ − ∂ w / ∂ x ) , the expressions of the bending moment and shear force can be obtained as: M = − E I ( k 1 2 a 1 e i k 1 x + k 1 2 d 1 e − i k 1 x + k 2 2 a 2 e i k 2 x + k 2 2 d 2 e − i k 2 x ) (8) V = κ G A [ ( g 1 − i k 1 ) a 1 e i k 1 x − ( g 1 − i k 1 ) d 1 e − i k 1 x + ( g 2 − i k 2 ) a 2 e i k 2 x − ( g 2 − i k 2 ) d 2 e − i k 2 x ] (9)

The Eqs 4, 5 can be rewritten in matrix form as W d = A d P h ( − x ) a + D d P h ( x ) d (10) and similarly, the Eqs 7, 8 are rewritten in matrix form as W f = A f P h ( − x ) a + D f P h ( x ) d (11) where W _d = {w, φ} ^T and W _f = {V, M} ^T are the generalized displacement vector and the generalized force vector, a and d respectively stand for the amplitude vectors of arriving wave and leaving wave, A _d and D _d respectively denote the coefficient matrixes of the arriving wave and the leaving wave corresponding to the displacement vector, A _f and D _f are the coefficient matrixes of the arriving wave and leaving wave corresponding to the force vector, and P _h denotes the phase matrix, which the specific expressions are given as follows: W d = { w φ } T W f = { V M } T (12) a = { a 1 a 2 } T d = { d 1 d 2 } T (13) P h ( x ) = [ e − i k 1 x 0 0 e − i k 2 x ] (14) A d = [ 1 1 g 1 g 2 ] D d = [ 1 1 − g 1 − g 2 ] (15) A f = [ κ G A 0 0 − E I ] [ g 1 − i k 1 g 2 − i k 2 k 1 2 k 2 2 ] D f = [ κ G A 0 0 − E I ] [ i k 1 − g 1 i k 2 − g 2 k 1 2 k 2 2 ] (16)

As the beam-oscillators coupling system in a unit cell plotted in Figure 1C , the relationships of the generalized displacements continuities and generalized forces equilibriums at the node J of the beam-oscillators coupling system unit cell are expressed as follows: W d J I = T d J W d J K (17) W f J I = T f J W f J K + F J (18) where T ^J _d =diag {-1, 1} and T _f ^J =-T ^J _d respectively stand for the transformation matrix of the generalized displacement and generalized force at node j, and F ^J denotes the reaction force vector of the local oscillator acting on the beam.

As the beam-oscillators coupling system in a unit cell illustrated in Figure 1C, the coupling vibration equation of the local oscillator coupled with the beam in parallel connection at the connection node can be obtained by m s w ¨ s + k s w s + c s w ˙ s = F J (19)

Therefore, the reaction force vector of the local oscillator acting on the beam is presented as F J = K J W d J K (20) where K ^J = diag{k _w k _φ } stand for the dynamic stiffness matrix of the local oscillator acting on the beam at node J, where the k _w = −m _s ω ²+k _s + ic _s ω and k _φ = 0 are the translational stiffness coefficient and rotational stiffness coefficient of the local oscillator acting on the beam, respectively, k _s and m _s respectively represent the spring stiffness and mass of the local oscillator, w _s = w ^JK = −w ^JI denotes the vertical displacement of the local oscillator and the beam at the node J.

Substituting Eqs 9, 10 and 19 into Eqs 16, 17 obtains the scattering relationship of the beam-oscillators coupling system at node J as follows: A J a J + D J d J = 0 (21) where a ^J = {( a ^JI ) ^T ( a ^JK ) ^T } ^T and d ^J = {( d ^JI ) ^T ( d ^JK ) ^T } ^T are the amplitude vectors of the arriving wave and the leaving wave, respectively, A ^J and D ^J respectively stand for the corresponding coefficient matrixes of a ^J and d ^J at the node J, respectively, which are expressed as A J = [ A d J I − T d J A d J K A f J I − ( T f J A f J K + K J A d J K ) ] D J = [ D d J I − T d J D d J K D f J I − ( T f J D f J K + K J D d J K ) ] (22)

According to the Bloch theorem of periodic structure, the following equations must be satisfied for the generalized displacement vector and generalized force vector in the unit cell of the proposed infinite long homogeneous beam coupled with periodic oscillators, which are listed as follows: e i q a W d I J = T d J W d K J (23) e i q a W f I J = T f J W f K J (24)

As the same, substituting Eqs 10, 11 into Eqs 22, 23 obtains A ∗ J a ∗ J + D ∗ J d ∗ J = 0 (25) where a * ^J = {( a ^IJ ) ^T ( a ^KJ ) ^T } ^T and d * ^J = {( d ^IJ ) ^T ( d ^KJ ) ^T } ^T are the amplitude vectors of the arriving wave and the leaving wave, respectively, A * ^J and D * ^J respectively stand for the corresponding coefficient matrixes of a * ^J and d * ^J at the node J, which are expressed as A ∗ J = [ e i q a A d I J − T d J A d K J e i q a A f I J − T f J A f K J ] D ∗ J = [ e i q a D d I J − T d J D d K J e i q a D f I J − T f J D f K J ] (26)

By combining Eqs 20, 24, the global scattering relationship in a unit cell of the infinite long homogeneous beam coupled with periodic oscillators can be obtained as follows: A a + D d = 0 (27) where a = {( a ^IJ ) ^T ( a ^JI ) ^T ( a ^JK ) ^T ( a ^KJ ) ^T } ^T and d = {( d ^IJ ) ^T ( d ^JI ) ^T ( d ^JK ) ^T ( d ^KJ ) ^T } ^T respectively stand for the global amplitude vectors of the arriving wave and the leaving wave of the proposed unit cell, A and D respectively denoted the corresponding coefficient matrixes of a and d at the node J, which the expressions are expressed as follows: A = [ e i q a A d I J 0 0 − T d J A d K J 0 A d J I − T d J A d J K 0 0 A f J I − ( T f J A f J K + K J A d J K ) 0 e i q a A f I J 0 0 − T f J A f K J ] D = [ e i q a D d I J 0 0 − T d J D d K J 0 D d J I − T d J D d J K 0 0 D f J I − ( T f J D f J K + K J D d J K ) 0 e i q a D f I J 0 0 − T f J D f K J ] (28)

For the flexural wave in any section of the beam, the leaving wave of the left end is exactly the arriving wave of the right node, and vice versa. For any section (e.g., section J) of the beam, the phase relationships can be expressed as follows: a J K = P J K d K J (29) a K J = P J K d J K (30) where P ^JK = − P _h (L ^JK ) is the phase matrix of the beam section J.

Thus, according to the phase relationships of each section of the beam, the global phase relationship is obtained as a = P d ∗ (31) where d ∗ = {( d ^JI ) ^T ( d ^IJ ) ^T ( d ^KJ ) ^T ( d ^JK ) ^T } ^T is the rearranged global amplitude vector of the leaving wave in the unit cell of the infinite long homogeneous beam coupled with periodic oscillators, P = diag{ P ^IJ P ^IJ P ^JK P ^JK } is the global phase matrix.

Comparing the global leaving wave amplitude vectors d ∗ and d of the infinite long homogeneous beam coupled with periodic oscillators unit cell, the two vectors have the same elements with different arrangement orders. Thus, the relationship between the amplitude vectors d ∗ and d is obtained as follows: d ∗ = U d (32) U = [ 0 I 2 0 0 I 2 0 0 0 0 0 0 I 2 0 0 I 2 0 ] (33) where U is the permutation matrix between d ∗ and d , in which I ₂ stands for the two-order unit matrix.

After substituting Eqs 30, 31 into Eq. 26, the flexural wave dispersion relation in the unit cell of the infinite long homogeneous beam coupled with periodic oscillators can be yielded as follows: ( A P U + D ) d = 0 (34) where R = APU + D denotes the reverberation-ray matrix of the periodic beam-oscillators coupling system.

In order to obtain the non-zero solution of the global leaving wave amplitude vector d , namely, the determinant of the reverberation-ray matrix R must be zero, which is det ( R ) = 0 (35)

Thus, the relationship of the flexural wave number q and frequency f in the unit cell of the proposed infinite long homogeneous beam coupled with periodic oscillators can be solved by Eq. 34, which is the flexural wave dispersion relations and vibration band gaps of the proposed periodic beam-oscillators coupling system.

In this paper, we propose a periodic beam-oscillators coupling system as the simple physical model for analyzing the dynamic characteristics of periodic support beams and low-frequency flexural wave vibration of slender stiffened plate structures in ship and offshore structures, the calculation method for the flexural wave band gaps in the proposed infinite long homogeneous beam coupled with periodic oscillators is established by the method of reverberation-ray matrix combined with Bloch theorem. The effectiveness of the flexural wave band gaps theoretical calculation method has been verified by the numerical results expressed by the flexural wave vibration transmission spectrum. Parametric studies show that the flexural wave band gaps can be adjusted and optimized manually by adjusting structural and material parameters. The studies are of great guiding significance for the vibration and noise control design of periodic support beams and slender stiffened plate structures, such as the 1D periodic structures in high-speed railway, building foundations, bridge structures, ships and offshore structures, and the 2D periodic stiffened plates in bridge structures, ships and offshore structures. The main conclusions are as follows:

1) The formation mechanism of the first band gap, which is determined by the local resonance of the periodic oscillators, while the second band gap is determined by the coupling effects between the Bragg scattering of periodic oscillators and the propagation of the flexural wave in homogeneous beam. As the lower limit frequency of the first flexural wave band gap keeps constant at 0 Hz, it is conducive to the application of the flexural wave band gap characteristics to the low-frequency vibration and noise control of the periodic structure.