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Flexural wave band gaps and vibration attenuation characteristics in periodic bi-directionally orthogonal stiffened plates
Ocean Engineering 178 (2019) 95–103
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Flexural wave band gaps and vibration attenuation characteristics in periodic bi-directionally orthogonal stiffened plates
T
Yinggang Lia,c,∗, Qingwen Zhoub, Lei Zhoub, Ling Zhub,c, Kailing Guob a
Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan, 430063, PR China Departments of Naval Architecture, Ocean and Structural Engineering, School of Transportation, Wuhan University of Technology, Wuhan, 430063, PR China c Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Wuhan, 430063, PR China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Periodic bi-directionally orthogonal stiffened plates Flexural wave band gaps Vibration attenuation Experimental validation
Vibrations in ship and offshore structures owing to various ocean environmental loads and excitations of power systems become increasingly serious. In this paper, the flexural wave propagation and vibration attenuation characteristics in periodic bi-directionally orthogonal stiffened plates are investigated. The dispersion relations and the displacement fields of the eigenmodes of infinite periodic bi-directionally orthogonal stiffened plates are calculated by using the finite element method in combination with Bloch periodic boundary conditions. Numerical results show that periodic bi-directionally orthogonal stiffened plates can yield complete and directional flexural wave band gaps, in which the propagation of flexural vibrational waves is prohibited and flexural vibration suppression is dramatically achieved. With the introduction of bi-directionally orthogonal stiffeners, the flexural wave and vibration energy is confined in the four corners of the plate owing to the scattering effect of the bi-directionally orthogonal stiffeners. The transmission spectra for a finite periodic stiffened plate are numerically and experimentally achieved to verify the existence of the flexural wave band gaps and vibration suppression characteristics. Furthermore, the effects of geometrical parameters on the flexural wave vibration band gaps are carried out. The flexural wave band gaps and vibration attenuation properties can be artificially modulated by changing the geometrical parameters of periodic bi-directionally orthogonal stiffened plates.
1. Introduction With the development of vessels with high-speed, large-scale and heavy-duty, ship and offshore structural vibrations owing to various ocean environmental loads reported by Meylan et al. (2015) and excitations of power systems investigated by Murawski and Charchalis (2014) become increasingly outstanding. Vibrations in ship and offshore structures may not only reduce platform productivity as well as serviceability and reliability of the structures but also have significantly adverse influences on the sound quality and comfort. Consequently, how to effectively and efficiently control vibration and noise of ship and offshore structures and realize low-noise design of structures appears extremely important. In an attempt to control vibration and noise of ship and offshore structures, a large number of studies have been carried out. Traditional vibration control strategies in ship and offshore structures are investigated and summarized by Lin et al. (2009), Kandasamy et al.
(2016) and Zhang et al. (2017). They can generally be divided into three categories, namely passive control method studied by Patil and Jangid (2005), Wang et al. (2013) and Yue et al. (2009) and semi-active control method investigated by Pinkaew and Fujino (2001), Wang and Li (2013) and Taghikhany et al. (2013) as well as active control method performed by Luo and Zhu (2006), Zhang et al. (2016) and Cinquemani and Braghin (2017). Passive methods are exemplified by the structural addition of (typically) viscoelastic damping layers, where the hysteresis loop of the cyclic stress and strain of the damping layer dissipates vibration energy. Semi-active control methods modify or control the mechanical properties of the damping element within the vibration control device. This usually requires a small amount of external power for operation (on the order of tens of watts). An active control typically requires a large power source for the operation of electrohydraulic or electromechanical (servo motor) actuator, which increases the structural damping or stiffness. Although the existing methods can be applied to control vibration and noise of ship and offshore structures in a
∗ Corresponding author. Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, PR China. E-mail address: [email protected] (Y. Li).
https://doi.org/10.1016/j.oceaneng.2019.02.076 Received 17 August 2018; Received in revised form 21 January 2019; Accepted 27 February 2019 0029-8018/ © 2019 Elsevier Ltd. All rights reserved.
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reasonable degree, they have some shortcomings, including temperature sensitivity and aging characteristics of viscoelastic damping materials in passive control, system complexity and narrow band characteristics of active control. In addition to the above traditional vibration control methods, a great deal of research has been conducted by Mead (1996), Achenbach (2012) and Trainiti et al. (2015) to explore the elastic wave propagation and attenuation properties in periodic structures over last few decades. Periodic structures can possess various unique physical properties; in particular, the existence of band gaps, in which the propagation of vibrational waves is prohibited. Yu et al. (2006) and Xiao et al. (2013) theoretically and experimentally studied the flexural wave propagation and vibration band gaps in one-dimensional Timoshenko beams with periodically attached local resonators. They demonstrated that the flexural vibration in one-dimensional beam structures can be suppressed significantly. Nobrega et al. (2016) numerically and experimentally investigated the elastic wave propagation and band gaps in one-dimensional elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators. Results showed that significant longitudinal vibration attenuation in one-dimensional periodic rods occurs. Richards and Pines (2003) designed a one-dimensional periodic shaft structure and showed that the proposed onedimensional periodic shaft can generate stop band and pass band regions in the frequency spectra, which was efficiently used to reduce transmitted vibration generated by gear mesh contact dynamics. Li et al. (2014) theoretically investigated the Lamb wave propagation and band gaps in one-dimensional radial periodic structures with periodic corrugation, which was utilized to control the radial vibration and radiated noise of rotational machines. Zhou et al. (2014) theoretically investigated the flexural wave propagation characteristics in one-dimensional periodically stiffened-thin-plate by using the center finite difference method. Results showed that one-dimensional periodically stiffened-thin-plate can yield complete band gaps and the structural vibration wave in the one-dimensional periodically stiffened-thin-plate is forbidden. Cho et al. (2016) presented free vibration numerical analysis of one-dimensional stiffened panels with multiple lumped mass and stiffness attachments. The Mindlin's theory was applied for plate and Timoshenko beam theory for stiffeners. Obviously, current researches on periodic structures are mainly concentrated on the wave propagation and vibration band gap characteristics in one-dimensional periodic structures such as periodic rods, beams and shafts as well as one-dimensional periodic plate structures, only structural vibration along periodic direction can be significantly isolated, whereas the structural vibration along non-periodic direction is difficult to control. Consequently, it is of considerable importance to investigate the elastic wave propagation and band gap properties in two-dimensional periodic plate structures for the two-dimensional structural vibration control in ship and offshore structures. Bi-directionally orthogonal stiffened plates are basic constitutive members of ships and offshore structures, which are widely used in deck and hull bottom as well as broadside structures to satisfy the requirement of structural safety and lightweight investigated by Mace (1981), Paik et al. (2001) and Chen and Xie (2005). In this paper, the flexural wave propagation and vibration attenuation characteristics in two-dimensional periodic bi-directionally orthogonal stiffened plates are numerically and experimentally investigated. Results show that periodic bidirectionally orthogonal stiffened plates can yield complete flexural wave band gaps, resulting in the significant flexural vibration suppression.
Fig. 1. Schematic of an 8 × 4 periodic bi-directionally orthogonal stiffened plate with lattice constant a0 = 120 mm, plate thickness e0 = 2 mm, stiffener height h0 = 20 mm and stiffener thickness b0 = 2 mm.
orthogonal stiffened plate. The lattice constant of the proposed stiffened plate is defined as a0. The infinite periodic bi-directionally orthogonal stiffened plate can be obtained by periodically repeating the unit cell along length direction and breadth direction. In this paper, the proposed finite periodic bi-directionally orthogonal stiffened plate is composed of 8 unit cells along the length direction and 4 unit cells along the breadth direction respectively. The lattice constant a0 of periodic bi-directionally orthogonal stiffened plate is equal to 120 mm. The overall dimensions of the proposed finite periodic stiffened plate are 960 mm × 480 mm with plate thickness e0 = 2 mm. The stiffener height and thickness are defined as h0 = 20 mm and b0 = 2 mm. In order to investigate the flexural wave propagation and vibration band gaps in the proposed periodic bi-directionally orthogonal stiffened plate, an efficient finite element method combination with Bloch theory is applied to calculate the dispersion relations and transmission power spectra. For the wave propagation in solid structures, the governing equation is given as follows: 3
∂uj ⎞ ⎤ ⎫ ∂ ⎡ ⎛ ∂ui ∂ 2u ⎧ ∂ ⎛ ∂uj ⎞ μ⎜ + = ρ 2i ⎜λ ⎟ + ⎟ ∂x j ⎢ ⎝ ∂x j ∂x i ⎠ ⎥ ⎬ ∂t i ⎝ ∂x j ⎠ ⎣ ⎦⎭ ⎩
∑ ⎨ ∂x j=1
i, j = x , y, z (1)
where u is the displacement vector, ρ is the mass density, t is the time, λ and μ are the Lame constants, x, y, and z represent the coordinate variables in Cartesian coordinate system respectively.
λ=
Eν E ,μ= (1 + ν )(1 − 2ν ) (1 + ν )
(2)
where E and ν represent the Young's modulus and Poisson's ratio respectively. Since the proposed bi-directionally orthogonal stiffened plate is typical thin-wall structure, the Structural Mechanics Module with Shell Application Mode in COMSOL Multiphysics is utilized to calculate the wave equation. The faces of the proposed stiffened plate are defined in the midplane. The dependent variables are the displacements u, v, and w in the global x, y, and z directions, and the rotations θx, θy, and θz about the global coordinate axes. The degrees of freedom defined by the shell element correspond to the values of the dependent variables in the three triangle vertices. For the calculation of the dispersion relations of the infinite periodic bi-directionally orthogonal stiffened plates, only the numerical calculation model of unit cell composed of bi-directionally orthogonal stiffeners deposited on a squad plate as shown in Fig. 2(a) should be considered since the infinite periodic system is in periodic distribution along the x-direction (length direction) and y-direction (breadth direction) simultaneously based on the translational periodicity and spatial symmetry of the proposed periodic bi-directionally stiffened
2. Physics model and calculation method The physics model of stiffened plate considered in this paper is constituted of bi-directionally orthogonal stiffeners periodically deposited on the plate as illustrated in Fig. 1. The area surrounded by red dashed line in Fig. 1 is the unit cell of periodic bi-directionally 96
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Fig. 2. Numerical calculation model of periodic bi-directionally orthogonal stiffened plate: (a) The unit cell for dispersion relations with Bloch boundary conditions; (b) The Irreducible Brillouin Zone; (c) The 8 × 4 periodic stiffened plate for the calculation of transmission spectrum with sine sweep acceleration excitation vibrating in z-direction and propagating along x-direction.
ky are the coordinate components of the Bloch wave vector limited to the first Brillouin zone. In this analysis, displacements are assumed to be time-harmonic with an angular frequency ω. A finite element discretization based on the typical cell in the standard generalized eigenvalue problem formulation
plates. According to the lattice theory and energy-band theory in solidstate physics reported by Birkhoff (1940) and Papaconstantopoulos (1986), the lattice vectors R = n1a1+n2a2, where n1 and n2 are integers, a1 and a2 are coplanar vertical vectors. The shell is described by its thickness and the material properties. The thickness of periodic stiffened plate is defined as 2 mm. The periodic stiffened plates are made of mild steel. The Young's modulus E is defined as 200 GPa, the Poisson's ratio ν is defined as 0.3, the mass density ρ is 7850 kg/m3 and the damping effect is neglected. In addition, the element used for the shell application mode is defined as Argyris shell proposed by Argyris et al. (1997), which is a simple but sophisticated 3-node shear-deformable isotropic and composite flat shell element suitable for largescale linear and nonlinear engineering computations of thin and thick anisotropic plate and complex shell structures. Its stiffness matrix is based on 12 straining modes but essentially requires the computation of a sparse 9 by 9 matrix. The transverse shear deformation is taken into account and the membrane and bending actions are uncoupled. The membrane action is modeled by a constant-strain triangle with true drilling rotations. The bending action is modeled by the bending part of Argyris shell element. Furthermore, stress-free boundary conditions are defined for free surfaces, while periodic boundary conditions with the Bloch theorem are applied at the boundaries as follows:
ui (x + a, y + a) = ui (x , y ) e j (kx a + ky a)
i = x, y
(K − ω2 M) u = 0
(4)
where K and M are the assembled stiffness and mass matrices for the typical cell. On the base of the Direct SPOOLES linear system solver, the eigenfrequency analysis can be conducted. Taking into account the spatial translation symmetry and point group symmetry of the infinite periodic system, only the Bloch wave vector along the boundary of the Irreducible Brillouin Zone as shown in Fig. 2(b) needs to be considered. The Brillouin zone is defined by the reciprocal lattice vector G = m1b1+m2b2, where m1 and m2 are integers. For a 2D periodic structure in a square configuration, the two base vectors are R1 (0, a) and R2 (a, 0); the corresponding reciprocal base vectors b1 and b2 can be expressed as b1=(2π/a, 0) and b2=(0, 2π/a). The periodic constant a is the size of a typical unit cell. With a given value of Bloch wave vector, a series of eigenfrequencies and eigenmodes can be obtained and the band structures of the proposed periodic stiffened plates can be obtained by repeating the eigenfrequency analysis. For the calculation of the transmission spectrum of periodic bi-directionally orthogonal stiffened plates, the finite array system
(3)
where x and y are the coordinate components of position vector; kx and 97
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Fig. 3. The band structure (a) and (b) transmission spectra of flexural wave in the periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2 mm, h0 = 20 mm and b0 = 2 mm.
flexural wave band gaps (960–1025 Hz and 1510–1680 Hz respectively in pink regions as shown Fig. 3(a)) along ΓX direction of the irreducible Brillouin zone occur in the band structure. Actually, the unit cell with periodic boundary is an ideal model which is applied to analyze the band gap characteristics, whereas the actual structures which are used in objective world are finite structures. In an attempt to validate the band structure of the infinite periodic stiffened plate and study the vibration attenuation performance in the frequency range of band gaps, the transmission power spectrum of 4 × 8 unit cells periodic bi-directionally orthogonal stiffened plate is carried out as shown in Fig. 3(b). One can observe that there are two obvious attenuation zones in the frequency range from 0 to 2000 Hz. The location and width of the attenuation zones in the transmission power spectrum is consistent with the flexural wave band gaps in the dispersion relation. It can be concluded that the flexural vibration wave propagation within specific frequency range is forbidden and dramatic flexural vibration wave attenuation could appear in the periodic bi-directionally orthogonal stiffened plate, resulting in the significant suppression of structural vibration. In addition, it can be also found that in other bands the periodic bi-directionally orthogonal stiffened plate enhances the spectrum (i.e. Transmission > 0 dB). This issue can be explained that we calculate the frequency response of the periodic bi-directionally orthogonal stiffened plates rather than the transmission spectrum and perfectly matched layers (PMLs) are not applied at the two ends of the finite plate in the x-direction to prevent the reflections. In order to intuitively elaborate the formation mechanism of the flexural wave band gaps in the periodic bi-directionally orthogonal stiffened plate, the total displacement fields of the unit cell in the specific wave vectors marked in Fig. 3(a) are conducted. Fig. 4(a–c) respectively illustrate the lower and upper edge eigenmodes of the complete flexural wave band gap as well as the upper edge eigenmodes of the first directional flexural wave band gap. It can be observed that the eigenmodes A and C are mainly embodied as the flexural wave propagation along x-direction and flexural vibration of the four corners in the plate and the bi-directionally orthogonal stiffeners remain
composed of 8 unit cells along the length direction and 4 unit cells along the breadth direction is considered as shown in Fig. 2(c). The acceleration excitation resource is defined as sine sweep excitation ranging from 0 to 2000 Hz with frequency interval 2 Hz. The vibration direction of the acceleration excitation resource is along z-direction and manifests as flexural vibration, while the propagation direction of the flexural wave is along x-direction. The arrows of acceleration excitation resource in Fig. 2(c) represent the vibration direction of the flexural wave. The plane wave acceleration sine excitation source with singlefrequency incidents from the left side of the finite array propagate along the x-direction. The transmitted acceleration value is detected and recorded on the right side of the structure. The transmission spectra are defined as the ratio of the transmitted power through the eight layered finite system to the incident power. By varying the excitation frequency of the incident waves, the transmission spectra can be obtained.
3. Results and discussions 3.1. The band gaps of periodic bi-directionally orthogonal stiffened plate Fig. 3(a) shows the calculated band structure of the proposed infinite periodic bi-directionally orthogonal stiffened plate by using the unit cell based on the finite element method. The left vertical axis is the wave frequency and the right vertical axis is the normalized frequency fa/cT, where f is the wave frequency and cT is the transversal wave speed of mild steel which is equal to 3140 m/s. The horizontal axis represents the reduced wave vector along the boundary of the irreducible Brillouin zone. Since the radiated noise from ship and offshore structures is mainly attributed to the flexural vibration of plate structures, the flexural wave propagation characteristics and vibration band gaps are our primary targets. We can find that there exist five bands in the frequency range from 0 to 2000 Hz including the traditional plate modes such as shear wave mode, symmetric and antisymmetric Lamb wave modes. Besides, one complete flexural wave band gap (710–960 Hz in cyan region as shown in Fig. 3(a)) and two directional 98
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Fig. 4. The total displacement fields of the eigenmodes of the corresponding point (A, B, C, D, E, F) marked in Fig. 3(a) and the flexural wave vibration propagation characteristics at specific frequencies for periodic stiffened plate.
band gaps in the periodic bi-directionally orthogonal stiffened plate, the flexural vibration transmission-measuring experiments of the finite array system composed of 8 unit cells along the length direction and 4 unit cells along the breadth direction were conducted. The manufactured structural model is shown in Fig. 5(a). The plate and the stiffeners were cut out from the mild steel plate by laser cutting to guarantee the machining accuracy. Afterwards, the plate and the stiffeners were connected by using superglue to guarantee the assembly accuracy and reduce initial deformation. Finally, the geometrical parameters of the manufactured model were determined as follows: the overall dimensions are 960 mm × 480 mm, the lattice constant is 120 mm, and the stiffener height is 20 mm. However, it can be found from Fig. 5(a) that the thickness of the manufactured stiffened plate is 2.4 mm, which is different from that of the numerical model because of the limitation of the raw material. In the flexural vibration transmission-measuring tests, the manufactured structural model was free-hanging by using soft ropes to achieve the free vibration property. M + P test system with a power amplifier and vibration exciter as well as two acceleration transducers was employed. The vibration exciter was utilized to perpendicularly stimulate the manufactured stiffened plate near one side. The experimental measurement implementation and the test setup are shown in Fig. 5(b) and (c). Then, the experimental measurements of the flexural wave attenuation and vibration suppression in the proposed stiffened plate were conducted. A white-noise random signal with bandwidth from 0 to 3000 Hz generated by M + P test system was applied to drive the exciter. Two B&K accelerometers were placed at two ends of the plate respectively. The one near the excitation point was used to detect the incident vibration wave signal while the other one was used to probe the transmitted vibration wave signal. Smart office software was applied to record experimental data and to obtain the experimental result
stationary, whereas the eigenmodes B is mainly attributed to the coupling between the flexural vibration of the plate and the torsional vibration of the bi-directionally orthogonal stiffeners. The eigenmodes D and F corresponding to the lower and upper edge of the second directional flexural wave band gap mainly manifest as the symmetric and antisymmetric flexural wave at the edge of plate along x-direction. Although eigenmodes E is located between eigenmodes D and F, the eigenmodes E is mainly embodied as the flexural wave at the edge of plate along y-direction, which has no effect on the second directional flexural wave band gap along x-direction. With the introduction of bidirectionally orthogonal stiffeners, the flexural wave and vibration energy is confined in the four corners of the plate owing to the scattering effect of the bi-directionally orthogonal stiffeners. In addition, in order to intuitively illustrate the suppression effect of the band gaps on the flexural wave vibration propagation, the wave propagation characteristics and the flexural vibration responses at specific frequencies for the finite periodic bi-directionally orthogonal stiffened plate are calculated as shown in Fig. 4(g) and (h). One can observe that the flexural wave vibration at 200 Hz which corresponds to the specific frequency outside the band gap can successfully propagate along the finite periodic bi-directionally orthogonal stiffened plate without significant attenuation. On the contrary, the flexural wave vibration at 970 Hz which corresponds to the specific frequency inside the band gap can hardly propagate along the finite periodic bi-directionally orthogonal stiffened plate, which means that the periodic bidirectionally orthogonal stiffened plate has a significant attenuation effect on the flexural wave vibration propagation. 3.2. Experimental validation To further demonstrate the existence of the flexural wave vibration 99
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Fig. 5. (a) The manufactured structural model of the finite array system; (b) The experimental measurement implementation; (c) Setup of the experimental measurement.
Fig. 6. The transmission spectra of the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm. The transmission spectra of experimental result and numerical result are illustrated in red solid dashedline and blue solid line respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
damping on the transmission spectrum is performed as shown in Fig. 6. Three different damping loss factors η are considered. It can be found that, with the introduction of material damping, significant attenuation of the transmission spectra at high frequency occur and the numerical results of the transmission spectra are consistent with the experimental result. Furthermore, the band structure for the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm is calculated as shown in Fig. 7. It can be observed that two flexural wave vibration band gaps (835–1198 Hz and 1575–2008 Hz respectively in pink regions) along ΓX direction of the irreducible Brillouin zone exist in the band structure of the manufactured structural model similarly. The location and width of
of the transmission power spectrum in the blue curve on the basis of spectral analysis and signal processing as shown in Fig. 6. Meanwhile, we perform the numerical calculation of the transmission spectrum for the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm in the red curve as shown in Fig. 6. One can observe from Fig. 6 that there are two significant vibration attenuation zones in the transmission spectrum by experimental test, which is in good agreement with numerical calculation result. The disagreement of the transmission value at high frequency between numerical simulation and experimental result is mainly attributed to the neglect of the superglue between the plate and the stiffeners as well as the material damping effects. The effect of 100
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Fig. 7. The numerical result of band structure of the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm.
Fig. 8. Gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of dimensionless lattice constant adim = a/a0, with e0 = 2 mm, h0 = 20 mm and b0 = 2 mm.
the stiffened plates and the geometrical parameters such as the plate thickness and stiffener thickness remain unchanged. Dimensionless lattice constant adim = a/a0 is defined to achieve dimensionless parameter study. The gap maps of the proposed periodic stiffened plates as a function of the factor of dimensionless lattice constant is illustrated in Fig. 8. One can observe that the dimensionless lattice constant has significant influence on the band gaps. With the increase of the dimensionless lattice constant from 0.5 to 3.0, the lower and upper edges of the band gaps shift to low-frequency range. This can be explained that the formation mechanism of the band gaps is mainly caused by the
the attenuation zones in the transmission spectra is consistent with the flexural wave band gaps in the dispersion relation.
3.3. Parametric study 3.3.1. Effect of lattice constant on the band gaps To investigate the effect of lattice constant on the flexural wave vibration band gaps, the dispersion relations of the periodic bi-directionally orthogonal stiffened plates with different lattice constant values are performed. During the calculation, the material parameters of 101
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Fig. 9. Gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of dimensionless plate thickness edim = e/a0, with a0 = 120 mm, h0 = 20 mm and b0 = 2 mm.
Fig. 10. Gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of dimensionless stiffener thickness bdim = b/a0, with a0 = 120 mm, e0 = 2 mm and h0 = 20 mm.
shown in Fig. 9. It can be found that, as the increase of the dimensionless plate thickness, the lower and upper edges of the flexural vibration band gaps move up to high frequency region. The bandwidth of the first band gap increases first and then remains at a certain level, whereas the bandwidth of the second band gap enlarges continually. This phenomenon is due to that the band gaps are resulted from the coupling effect between the flexural wave vibration of the plate and the Bragg scattering of the bi-directionally orthogonal stiffeners. With the increase of the dimensionless plate thickness, the bending stiffness of the stiffened plate enhances, resulting in the increase of the eigenfrequencies.
Bragg scattering of periodic bi-directionally orthogonal stiffeners on the flexural wave in the plate. According to the Bragg scattering theory, the wave lengths corresponding to the mid-frequency of the band gaps in periodic structures is approximately two times of the lattice constant. Consequently, the low-frequency vibration control in ship and offshore structures can be effectively achieved by tuning the dimensionless lattice constant of the stiffened plates.
3.3.2. Effect of plate thickness on the band gaps Keep the material parameters of the stiffened plates and the geometrical parameters such as the lattice constant and the thickness and height of stiffener, the effect of plate thickness on the flexural wave vibration band gaps of the periodic bi-directionally orthogonal stiffened plates are conducted. Dimensionless plate thickness edim = e/a0 is defined to achieve dimensionless parameter study. The gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of the factor of dimensionless plate thickness is obtained as
3.3.3. Effect of stiffener thickness on the band gaps In order to investigate the effect of stiffener thickness on the flexural wave vibration band gaps, the gap maps of the proposed periodic stiffened plates as a function of the factor of dimensionless stiffener thickness bdim is carried out as shown in Fig. 10. The dimensionless 102
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stiffener thickness bdim is defined as b/a0. It can be observed that, as the dimensionless stiffener thickness increases, the lower edge of the first and second band gaps goes up gradually and the upper edge of the band gap almost remains unchanged. This phenomenon is due to that the eigenmodes of the band edges are embodied as the flexural vibration of the stiffened plate. As the increase of the dimensionless stiffener thickness, the bending stiffness of the stiffened plate enhances, resulting in the increase of the eigenfrequencies.
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4. Concluding remarks In this paper, the flexural wave propagation and vibration suppression characteristics in periodic bi-directionally orthogonal stiffened plates are studied. A standard finite element method in combination with Bloch periodic boundary conditions is applied to solve the elastic wave equation and calculate the dispersion relations and the transmission spectra. Experimental measurement of the flexural vibration transmission spectrum for a finite periodic stiffened plate was conducted to validate the numerical calculation method. Results show that there exist complete and directional flexural wave band gaps of periodic bi-directionally orthogonal stiffened plates. The flexural vibration wave propagation within specific frequency range is forbidden and dramatic flexural vibration wave attenuation could appear in the periodic bidirectionally orthogonal stiffened plate, resulting in the significant suppression of structural vibration. Furthermore, the flexural wave vibration band gaps can be artificially modulated and optimized by tuning the geometrical parameters. Acknowledgements The authors gratefully acknowledge financial support from the project of National Natural Science Foundation of China (No. 11602182) and the Open Fund of Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education (GXNC18041402). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.oceaneng.2019.02.076. References Achenbach, J., 2012. Wave Propagation in Elastic Solids. Elsevier. Argyris, J., Tenek, L., Olofsson, L., 1997. TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells. Comput. Methods Appl. Mech. Eng. 145 (1–2), 11–85. Birkhoff, G., 1940. Lattice Theory. American Mathematical Soc. Chen, Z., Xie, W.C., 2005. Vibration localization in plates rib-stiffened in two orthogonal
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Flexural wave band gaps and vibration attenuation characteristics in periodic bi-directionally orthogonal stiffened plates
T
Yinggang Lia,c,∗, Qingwen Zhoub, Lei Zhoub, Ling Zhub,c, Kailing Guob a
Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan, 430063, PR China Departments of Naval Architecture, Ocean and Structural Engineering, School of Transportation, Wuhan University of Technology, Wuhan, 430063, PR China c Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Wuhan, 430063, PR China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Periodic bi-directionally orthogonal stiffened plates Flexural wave band gaps Vibration attenuation Experimental validation
Vibrations in ship and offshore structures owing to various ocean environmental loads and excitations of power systems become increasingly serious. In this paper, the flexural wave propagation and vibration attenuation characteristics in periodic bi-directionally orthogonal stiffened plates are investigated. The dispersion relations and the displacement fields of the eigenmodes of infinite periodic bi-directionally orthogonal stiffened plates are calculated by using the finite element method in combination with Bloch periodic boundary conditions. Numerical results show that periodic bi-directionally orthogonal stiffened plates can yield complete and directional flexural wave band gaps, in which the propagation of flexural vibrational waves is prohibited and flexural vibration suppression is dramatically achieved. With the introduction of bi-directionally orthogonal stiffeners, the flexural wave and vibration energy is confined in the four corners of the plate owing to the scattering effect of the bi-directionally orthogonal stiffeners. The transmission spectra for a finite periodic stiffened plate are numerically and experimentally achieved to verify the existence of the flexural wave band gaps and vibration suppression characteristics. Furthermore, the effects of geometrical parameters on the flexural wave vibration band gaps are carried out. The flexural wave band gaps and vibration attenuation properties can be artificially modulated by changing the geometrical parameters of periodic bi-directionally orthogonal stiffened plates.
1. Introduction With the development of vessels with high-speed, large-scale and heavy-duty, ship and offshore structural vibrations owing to various ocean environmental loads reported by Meylan et al. (2015) and excitations of power systems investigated by Murawski and Charchalis (2014) become increasingly outstanding. Vibrations in ship and offshore structures may not only reduce platform productivity as well as serviceability and reliability of the structures but also have significantly adverse influences on the sound quality and comfort. Consequently, how to effectively and efficiently control vibration and noise of ship and offshore structures and realize low-noise design of structures appears extremely important. In an attempt to control vibration and noise of ship and offshore structures, a large number of studies have been carried out. Traditional vibration control strategies in ship and offshore structures are investigated and summarized by Lin et al. (2009), Kandasamy et al.
(2016) and Zhang et al. (2017). They can generally be divided into three categories, namely passive control method studied by Patil and Jangid (2005), Wang et al. (2013) and Yue et al. (2009) and semi-active control method investigated by Pinkaew and Fujino (2001), Wang and Li (2013) and Taghikhany et al. (2013) as well as active control method performed by Luo and Zhu (2006), Zhang et al. (2016) and Cinquemani and Braghin (2017). Passive methods are exemplified by the structural addition of (typically) viscoelastic damping layers, where the hysteresis loop of the cyclic stress and strain of the damping layer dissipates vibration energy. Semi-active control methods modify or control the mechanical properties of the damping element within the vibration control device. This usually requires a small amount of external power for operation (on the order of tens of watts). An active control typically requires a large power source for the operation of electrohydraulic or electromechanical (servo motor) actuator, which increases the structural damping or stiffness. Although the existing methods can be applied to control vibration and noise of ship and offshore structures in a
∗ Corresponding author. Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, PR China. E-mail address: [email protected] (Y. Li).
https://doi.org/10.1016/j.oceaneng.2019.02.076 Received 17 August 2018; Received in revised form 21 January 2019; Accepted 27 February 2019 0029-8018/ © 2019 Elsevier Ltd. All rights reserved.
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reasonable degree, they have some shortcomings, including temperature sensitivity and aging characteristics of viscoelastic damping materials in passive control, system complexity and narrow band characteristics of active control. In addition to the above traditional vibration control methods, a great deal of research has been conducted by Mead (1996), Achenbach (2012) and Trainiti et al. (2015) to explore the elastic wave propagation and attenuation properties in periodic structures over last few decades. Periodic structures can possess various unique physical properties; in particular, the existence of band gaps, in which the propagation of vibrational waves is prohibited. Yu et al. (2006) and Xiao et al. (2013) theoretically and experimentally studied the flexural wave propagation and vibration band gaps in one-dimensional Timoshenko beams with periodically attached local resonators. They demonstrated that the flexural vibration in one-dimensional beam structures can be suppressed significantly. Nobrega et al. (2016) numerically and experimentally investigated the elastic wave propagation and band gaps in one-dimensional elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators. Results showed that significant longitudinal vibration attenuation in one-dimensional periodic rods occurs. Richards and Pines (2003) designed a one-dimensional periodic shaft structure and showed that the proposed onedimensional periodic shaft can generate stop band and pass band regions in the frequency spectra, which was efficiently used to reduce transmitted vibration generated by gear mesh contact dynamics. Li et al. (2014) theoretically investigated the Lamb wave propagation and band gaps in one-dimensional radial periodic structures with periodic corrugation, which was utilized to control the radial vibration and radiated noise of rotational machines. Zhou et al. (2014) theoretically investigated the flexural wave propagation characteristics in one-dimensional periodically stiffened-thin-plate by using the center finite difference method. Results showed that one-dimensional periodically stiffened-thin-plate can yield complete band gaps and the structural vibration wave in the one-dimensional periodically stiffened-thin-plate is forbidden. Cho et al. (2016) presented free vibration numerical analysis of one-dimensional stiffened panels with multiple lumped mass and stiffness attachments. The Mindlin's theory was applied for plate and Timoshenko beam theory for stiffeners. Obviously, current researches on periodic structures are mainly concentrated on the wave propagation and vibration band gap characteristics in one-dimensional periodic structures such as periodic rods, beams and shafts as well as one-dimensional periodic plate structures, only structural vibration along periodic direction can be significantly isolated, whereas the structural vibration along non-periodic direction is difficult to control. Consequently, it is of considerable importance to investigate the elastic wave propagation and band gap properties in two-dimensional periodic plate structures for the two-dimensional structural vibration control in ship and offshore structures. Bi-directionally orthogonal stiffened plates are basic constitutive members of ships and offshore structures, which are widely used in deck and hull bottom as well as broadside structures to satisfy the requirement of structural safety and lightweight investigated by Mace (1981), Paik et al. (2001) and Chen and Xie (2005). In this paper, the flexural wave propagation and vibration attenuation characteristics in two-dimensional periodic bi-directionally orthogonal stiffened plates are numerically and experimentally investigated. Results show that periodic bidirectionally orthogonal stiffened plates can yield complete flexural wave band gaps, resulting in the significant flexural vibration suppression.
Fig. 1. Schematic of an 8 × 4 periodic bi-directionally orthogonal stiffened plate with lattice constant a0 = 120 mm, plate thickness e0 = 2 mm, stiffener height h0 = 20 mm and stiffener thickness b0 = 2 mm.
orthogonal stiffened plate. The lattice constant of the proposed stiffened plate is defined as a0. The infinite periodic bi-directionally orthogonal stiffened plate can be obtained by periodically repeating the unit cell along length direction and breadth direction. In this paper, the proposed finite periodic bi-directionally orthogonal stiffened plate is composed of 8 unit cells along the length direction and 4 unit cells along the breadth direction respectively. The lattice constant a0 of periodic bi-directionally orthogonal stiffened plate is equal to 120 mm. The overall dimensions of the proposed finite periodic stiffened plate are 960 mm × 480 mm with plate thickness e0 = 2 mm. The stiffener height and thickness are defined as h0 = 20 mm and b0 = 2 mm. In order to investigate the flexural wave propagation and vibration band gaps in the proposed periodic bi-directionally orthogonal stiffened plate, an efficient finite element method combination with Bloch theory is applied to calculate the dispersion relations and transmission power spectra. For the wave propagation in solid structures, the governing equation is given as follows: 3
∂uj ⎞ ⎤ ⎫ ∂ ⎡ ⎛ ∂ui ∂ 2u ⎧ ∂ ⎛ ∂uj ⎞ μ⎜ + = ρ 2i ⎜λ ⎟ + ⎟ ∂x j ⎢ ⎝ ∂x j ∂x i ⎠ ⎥ ⎬ ∂t i ⎝ ∂x j ⎠ ⎣ ⎦⎭ ⎩
∑ ⎨ ∂x j=1
i, j = x , y, z (1)
where u is the displacement vector, ρ is the mass density, t is the time, λ and μ are the Lame constants, x, y, and z represent the coordinate variables in Cartesian coordinate system respectively.
λ=
Eν E ,μ= (1 + ν )(1 − 2ν ) (1 + ν )
(2)
where E and ν represent the Young's modulus and Poisson's ratio respectively. Since the proposed bi-directionally orthogonal stiffened plate is typical thin-wall structure, the Structural Mechanics Module with Shell Application Mode in COMSOL Multiphysics is utilized to calculate the wave equation. The faces of the proposed stiffened plate are defined in the midplane. The dependent variables are the displacements u, v, and w in the global x, y, and z directions, and the rotations θx, θy, and θz about the global coordinate axes. The degrees of freedom defined by the shell element correspond to the values of the dependent variables in the three triangle vertices. For the calculation of the dispersion relations of the infinite periodic bi-directionally orthogonal stiffened plates, only the numerical calculation model of unit cell composed of bi-directionally orthogonal stiffeners deposited on a squad plate as shown in Fig. 2(a) should be considered since the infinite periodic system is in periodic distribution along the x-direction (length direction) and y-direction (breadth direction) simultaneously based on the translational periodicity and spatial symmetry of the proposed periodic bi-directionally stiffened
2. Physics model and calculation method The physics model of stiffened plate considered in this paper is constituted of bi-directionally orthogonal stiffeners periodically deposited on the plate as illustrated in Fig. 1. The area surrounded by red dashed line in Fig. 1 is the unit cell of periodic bi-directionally 96
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Fig. 2. Numerical calculation model of periodic bi-directionally orthogonal stiffened plate: (a) The unit cell for dispersion relations with Bloch boundary conditions; (b) The Irreducible Brillouin Zone; (c) The 8 × 4 periodic stiffened plate for the calculation of transmission spectrum with sine sweep acceleration excitation vibrating in z-direction and propagating along x-direction.
ky are the coordinate components of the Bloch wave vector limited to the first Brillouin zone. In this analysis, displacements are assumed to be time-harmonic with an angular frequency ω. A finite element discretization based on the typical cell in the standard generalized eigenvalue problem formulation
plates. According to the lattice theory and energy-band theory in solidstate physics reported by Birkhoff (1940) and Papaconstantopoulos (1986), the lattice vectors R = n1a1+n2a2, where n1 and n2 are integers, a1 and a2 are coplanar vertical vectors. The shell is described by its thickness and the material properties. The thickness of periodic stiffened plate is defined as 2 mm. The periodic stiffened plates are made of mild steel. The Young's modulus E is defined as 200 GPa, the Poisson's ratio ν is defined as 0.3, the mass density ρ is 7850 kg/m3 and the damping effect is neglected. In addition, the element used for the shell application mode is defined as Argyris shell proposed by Argyris et al. (1997), which is a simple but sophisticated 3-node shear-deformable isotropic and composite flat shell element suitable for largescale linear and nonlinear engineering computations of thin and thick anisotropic plate and complex shell structures. Its stiffness matrix is based on 12 straining modes but essentially requires the computation of a sparse 9 by 9 matrix. The transverse shear deformation is taken into account and the membrane and bending actions are uncoupled. The membrane action is modeled by a constant-strain triangle with true drilling rotations. The bending action is modeled by the bending part of Argyris shell element. Furthermore, stress-free boundary conditions are defined for free surfaces, while periodic boundary conditions with the Bloch theorem are applied at the boundaries as follows:
ui (x + a, y + a) = ui (x , y ) e j (kx a + ky a)
i = x, y
(K − ω2 M) u = 0
(4)
where K and M are the assembled stiffness and mass matrices for the typical cell. On the base of the Direct SPOOLES linear system solver, the eigenfrequency analysis can be conducted. Taking into account the spatial translation symmetry and point group symmetry of the infinite periodic system, only the Bloch wave vector along the boundary of the Irreducible Brillouin Zone as shown in Fig. 2(b) needs to be considered. The Brillouin zone is defined by the reciprocal lattice vector G = m1b1+m2b2, where m1 and m2 are integers. For a 2D periodic structure in a square configuration, the two base vectors are R1 (0, a) and R2 (a, 0); the corresponding reciprocal base vectors b1 and b2 can be expressed as b1=(2π/a, 0) and b2=(0, 2π/a). The periodic constant a is the size of a typical unit cell. With a given value of Bloch wave vector, a series of eigenfrequencies and eigenmodes can be obtained and the band structures of the proposed periodic stiffened plates can be obtained by repeating the eigenfrequency analysis. For the calculation of the transmission spectrum of periodic bi-directionally orthogonal stiffened plates, the finite array system
(3)
where x and y are the coordinate components of position vector; kx and 97
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Fig. 3. The band structure (a) and (b) transmission spectra of flexural wave in the periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2 mm, h0 = 20 mm and b0 = 2 mm.
flexural wave band gaps (960–1025 Hz and 1510–1680 Hz respectively in pink regions as shown Fig. 3(a)) along ΓX direction of the irreducible Brillouin zone occur in the band structure. Actually, the unit cell with periodic boundary is an ideal model which is applied to analyze the band gap characteristics, whereas the actual structures which are used in objective world are finite structures. In an attempt to validate the band structure of the infinite periodic stiffened plate and study the vibration attenuation performance in the frequency range of band gaps, the transmission power spectrum of 4 × 8 unit cells periodic bi-directionally orthogonal stiffened plate is carried out as shown in Fig. 3(b). One can observe that there are two obvious attenuation zones in the frequency range from 0 to 2000 Hz. The location and width of the attenuation zones in the transmission power spectrum is consistent with the flexural wave band gaps in the dispersion relation. It can be concluded that the flexural vibration wave propagation within specific frequency range is forbidden and dramatic flexural vibration wave attenuation could appear in the periodic bi-directionally orthogonal stiffened plate, resulting in the significant suppression of structural vibration. In addition, it can be also found that in other bands the periodic bi-directionally orthogonal stiffened plate enhances the spectrum (i.e. Transmission > 0 dB). This issue can be explained that we calculate the frequency response of the periodic bi-directionally orthogonal stiffened plates rather than the transmission spectrum and perfectly matched layers (PMLs) are not applied at the two ends of the finite plate in the x-direction to prevent the reflections. In order to intuitively elaborate the formation mechanism of the flexural wave band gaps in the periodic bi-directionally orthogonal stiffened plate, the total displacement fields of the unit cell in the specific wave vectors marked in Fig. 3(a) are conducted. Fig. 4(a–c) respectively illustrate the lower and upper edge eigenmodes of the complete flexural wave band gap as well as the upper edge eigenmodes of the first directional flexural wave band gap. It can be observed that the eigenmodes A and C are mainly embodied as the flexural wave propagation along x-direction and flexural vibration of the four corners in the plate and the bi-directionally orthogonal stiffeners remain
composed of 8 unit cells along the length direction and 4 unit cells along the breadth direction is considered as shown in Fig. 2(c). The acceleration excitation resource is defined as sine sweep excitation ranging from 0 to 2000 Hz with frequency interval 2 Hz. The vibration direction of the acceleration excitation resource is along z-direction and manifests as flexural vibration, while the propagation direction of the flexural wave is along x-direction. The arrows of acceleration excitation resource in Fig. 2(c) represent the vibration direction of the flexural wave. The plane wave acceleration sine excitation source with singlefrequency incidents from the left side of the finite array propagate along the x-direction. The transmitted acceleration value is detected and recorded on the right side of the structure. The transmission spectra are defined as the ratio of the transmitted power through the eight layered finite system to the incident power. By varying the excitation frequency of the incident waves, the transmission spectra can be obtained.
3. Results and discussions 3.1. The band gaps of periodic bi-directionally orthogonal stiffened plate Fig. 3(a) shows the calculated band structure of the proposed infinite periodic bi-directionally orthogonal stiffened plate by using the unit cell based on the finite element method. The left vertical axis is the wave frequency and the right vertical axis is the normalized frequency fa/cT, where f is the wave frequency and cT is the transversal wave speed of mild steel which is equal to 3140 m/s. The horizontal axis represents the reduced wave vector along the boundary of the irreducible Brillouin zone. Since the radiated noise from ship and offshore structures is mainly attributed to the flexural vibration of plate structures, the flexural wave propagation characteristics and vibration band gaps are our primary targets. We can find that there exist five bands in the frequency range from 0 to 2000 Hz including the traditional plate modes such as shear wave mode, symmetric and antisymmetric Lamb wave modes. Besides, one complete flexural wave band gap (710–960 Hz in cyan region as shown in Fig. 3(a)) and two directional 98
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Fig. 4. The total displacement fields of the eigenmodes of the corresponding point (A, B, C, D, E, F) marked in Fig. 3(a) and the flexural wave vibration propagation characteristics at specific frequencies for periodic stiffened plate.
band gaps in the periodic bi-directionally orthogonal stiffened plate, the flexural vibration transmission-measuring experiments of the finite array system composed of 8 unit cells along the length direction and 4 unit cells along the breadth direction were conducted. The manufactured structural model is shown in Fig. 5(a). The plate and the stiffeners were cut out from the mild steel plate by laser cutting to guarantee the machining accuracy. Afterwards, the plate and the stiffeners were connected by using superglue to guarantee the assembly accuracy and reduce initial deformation. Finally, the geometrical parameters of the manufactured model were determined as follows: the overall dimensions are 960 mm × 480 mm, the lattice constant is 120 mm, and the stiffener height is 20 mm. However, it can be found from Fig. 5(a) that the thickness of the manufactured stiffened plate is 2.4 mm, which is different from that of the numerical model because of the limitation of the raw material. In the flexural vibration transmission-measuring tests, the manufactured structural model was free-hanging by using soft ropes to achieve the free vibration property. M + P test system with a power amplifier and vibration exciter as well as two acceleration transducers was employed. The vibration exciter was utilized to perpendicularly stimulate the manufactured stiffened plate near one side. The experimental measurement implementation and the test setup are shown in Fig. 5(b) and (c). Then, the experimental measurements of the flexural wave attenuation and vibration suppression in the proposed stiffened plate were conducted. A white-noise random signal with bandwidth from 0 to 3000 Hz generated by M + P test system was applied to drive the exciter. Two B&K accelerometers were placed at two ends of the plate respectively. The one near the excitation point was used to detect the incident vibration wave signal while the other one was used to probe the transmitted vibration wave signal. Smart office software was applied to record experimental data and to obtain the experimental result
stationary, whereas the eigenmodes B is mainly attributed to the coupling between the flexural vibration of the plate and the torsional vibration of the bi-directionally orthogonal stiffeners. The eigenmodes D and F corresponding to the lower and upper edge of the second directional flexural wave band gap mainly manifest as the symmetric and antisymmetric flexural wave at the edge of plate along x-direction. Although eigenmodes E is located between eigenmodes D and F, the eigenmodes E is mainly embodied as the flexural wave at the edge of plate along y-direction, which has no effect on the second directional flexural wave band gap along x-direction. With the introduction of bidirectionally orthogonal stiffeners, the flexural wave and vibration energy is confined in the four corners of the plate owing to the scattering effect of the bi-directionally orthogonal stiffeners. In addition, in order to intuitively illustrate the suppression effect of the band gaps on the flexural wave vibration propagation, the wave propagation characteristics and the flexural vibration responses at specific frequencies for the finite periodic bi-directionally orthogonal stiffened plate are calculated as shown in Fig. 4(g) and (h). One can observe that the flexural wave vibration at 200 Hz which corresponds to the specific frequency outside the band gap can successfully propagate along the finite periodic bi-directionally orthogonal stiffened plate without significant attenuation. On the contrary, the flexural wave vibration at 970 Hz which corresponds to the specific frequency inside the band gap can hardly propagate along the finite periodic bi-directionally orthogonal stiffened plate, which means that the periodic bidirectionally orthogonal stiffened plate has a significant attenuation effect on the flexural wave vibration propagation. 3.2. Experimental validation To further demonstrate the existence of the flexural wave vibration 99
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Fig. 5. (a) The manufactured structural model of the finite array system; (b) The experimental measurement implementation; (c) Setup of the experimental measurement.
Fig. 6. The transmission spectra of the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm. The transmission spectra of experimental result and numerical result are illustrated in red solid dashedline and blue solid line respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
damping on the transmission spectrum is performed as shown in Fig. 6. Three different damping loss factors η are considered. It can be found that, with the introduction of material damping, significant attenuation of the transmission spectra at high frequency occur and the numerical results of the transmission spectra are consistent with the experimental result. Furthermore, the band structure for the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm is calculated as shown in Fig. 7. It can be observed that two flexural wave vibration band gaps (835–1198 Hz and 1575–2008 Hz respectively in pink regions) along ΓX direction of the irreducible Brillouin zone exist in the band structure of the manufactured structural model similarly. The location and width of
of the transmission power spectrum in the blue curve on the basis of spectral analysis and signal processing as shown in Fig. 6. Meanwhile, we perform the numerical calculation of the transmission spectrum for the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm in the red curve as shown in Fig. 6. One can observe from Fig. 6 that there are two significant vibration attenuation zones in the transmission spectrum by experimental test, which is in good agreement with numerical calculation result. The disagreement of the transmission value at high frequency between numerical simulation and experimental result is mainly attributed to the neglect of the superglue between the plate and the stiffeners as well as the material damping effects. The effect of 100
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Fig. 7. The numerical result of band structure of the manufactured periodic bi-directionally orthogonal stiffened plate with a0 = 120 mm, e0 = 2.4 mm, h0 = 20 mm and b0 = 2.4 mm.
Fig. 8. Gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of dimensionless lattice constant adim = a/a0, with e0 = 2 mm, h0 = 20 mm and b0 = 2 mm.
the stiffened plates and the geometrical parameters such as the plate thickness and stiffener thickness remain unchanged. Dimensionless lattice constant adim = a/a0 is defined to achieve dimensionless parameter study. The gap maps of the proposed periodic stiffened plates as a function of the factor of dimensionless lattice constant is illustrated in Fig. 8. One can observe that the dimensionless lattice constant has significant influence on the band gaps. With the increase of the dimensionless lattice constant from 0.5 to 3.0, the lower and upper edges of the band gaps shift to low-frequency range. This can be explained that the formation mechanism of the band gaps is mainly caused by the
the attenuation zones in the transmission spectra is consistent with the flexural wave band gaps in the dispersion relation.
3.3. Parametric study 3.3.1. Effect of lattice constant on the band gaps To investigate the effect of lattice constant on the flexural wave vibration band gaps, the dispersion relations of the periodic bi-directionally orthogonal stiffened plates with different lattice constant values are performed. During the calculation, the material parameters of 101
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Fig. 9. Gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of dimensionless plate thickness edim = e/a0, with a0 = 120 mm, h0 = 20 mm and b0 = 2 mm.
Fig. 10. Gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of dimensionless stiffener thickness bdim = b/a0, with a0 = 120 mm, e0 = 2 mm and h0 = 20 mm.
shown in Fig. 9. It can be found that, as the increase of the dimensionless plate thickness, the lower and upper edges of the flexural vibration band gaps move up to high frequency region. The bandwidth of the first band gap increases first and then remains at a certain level, whereas the bandwidth of the second band gap enlarges continually. This phenomenon is due to that the band gaps are resulted from the coupling effect between the flexural wave vibration of the plate and the Bragg scattering of the bi-directionally orthogonal stiffeners. With the increase of the dimensionless plate thickness, the bending stiffness of the stiffened plate enhances, resulting in the increase of the eigenfrequencies.
Bragg scattering of periodic bi-directionally orthogonal stiffeners on the flexural wave in the plate. According to the Bragg scattering theory, the wave lengths corresponding to the mid-frequency of the band gaps in periodic structures is approximately two times of the lattice constant. Consequently, the low-frequency vibration control in ship and offshore structures can be effectively achieved by tuning the dimensionless lattice constant of the stiffened plates.
3.3.2. Effect of plate thickness on the band gaps Keep the material parameters of the stiffened plates and the geometrical parameters such as the lattice constant and the thickness and height of stiffener, the effect of plate thickness on the flexural wave vibration band gaps of the periodic bi-directionally orthogonal stiffened plates are conducted. Dimensionless plate thickness edim = e/a0 is defined to achieve dimensionless parameter study. The gap maps of the proposed periodic bi-directionally orthogonal stiffened plates as a function of the factor of dimensionless plate thickness is obtained as
3.3.3. Effect of stiffener thickness on the band gaps In order to investigate the effect of stiffener thickness on the flexural wave vibration band gaps, the gap maps of the proposed periodic stiffened plates as a function of the factor of dimensionless stiffener thickness bdim is carried out as shown in Fig. 10. The dimensionless 102
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stiffener thickness bdim is defined as b/a0. It can be observed that, as the dimensionless stiffener thickness increases, the lower edge of the first and second band gaps goes up gradually and the upper edge of the band gap almost remains unchanged. This phenomenon is due to that the eigenmodes of the band edges are embodied as the flexural vibration of the stiffened plate. As the increase of the dimensionless stiffener thickness, the bending stiffness of the stiffened plate enhances, resulting in the increase of the eigenfrequencies.
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4. Concluding remarks In this paper, the flexural wave propagation and vibration suppression characteristics in periodic bi-directionally orthogonal stiffened plates are studied. A standard finite element method in combination with Bloch periodic boundary conditions is applied to solve the elastic wave equation and calculate the dispersion relations and the transmission spectra. Experimental measurement of the flexural vibration transmission spectrum for a finite periodic stiffened plate was conducted to validate the numerical calculation method. Results show that there exist complete and directional flexural wave band gaps of periodic bi-directionally orthogonal stiffened plates. The flexural vibration wave propagation within specific frequency range is forbidden and dramatic flexural vibration wave attenuation could appear in the periodic bidirectionally orthogonal stiffened plate, resulting in the significant suppression of structural vibration. Furthermore, the flexural wave vibration band gaps can be artificially modulated and optimized by tuning the geometrical parameters. Acknowledgements The authors gratefully acknowledge financial support from the project of National Natural Science Foundation of China (No. 11602182) and the Open Fund of Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education (GXNC18041402). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.oceaneng.2019.02.076. References Achenbach, J., 2012. Wave Propagation in Elastic Solids. Elsevier. Argyris, J., Tenek, L., Olofsson, L., 1997. TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells. Comput. Methods Appl. Mech. Eng. 145 (1–2), 11–85. Birkhoff, G., 1940. Lattice Theory. American Mathematical Soc. Chen, Z., Xie, W.C., 2005. Vibration localization in plates rib-stiffened in two orthogonal
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