Geometry and material parameters of the plates (case 1).
Abstract
Initiated by the objective to address the dynamics of ship structures other than conventional finite element method, a dynamic stiffness method (DSM) is proposed systematically including that for three types of element models. A DSM element accounting for both inplane and bending vibrations in flat rectangular plates is developed, which makes it possible for modeling wave conversion across junctions in builtup plates. In addition, a DSM element for stiffened plates is formulated, which considers all possible vibrations in plates and beams, i.e., bending, torsion, and extension motions. The third type of DSM plate element takes fluid loading into account, which is induced by vibrating plate. Finally, the proposed DSM method is extended to address vibration transmission in a builtup plate structure, which demonstrates the great potentials of DSM in application to more practical and more general engineering fields.
Keywords
 dynamic stiffness method
 FEM
 power flow
 beamstiffened
 ship structures
1. Introduction
The vibrational and acoustic characteristics of ship structures are likely to be one of a number of practical concerns not only to mechanical designers and research scientists, but sometimes even to military defense officers. The reasons lie in the following facts. Firstly, excessive vibration levels induced by operating machines or incident waves can inevitably lead to structural fatigue, failure, or even unexpected disasters. Besides, onboard vibration and noise are one of the most important indexes on ship habitability. Too much exposure to such vibrational and noisy environments can make ship crew members uncomfortable, fatigue, or even unhealthy, which has been convinced from a lot of experimental data, and even witnessed from many ship collision accidents. Last but not least, for naval ships, their vibration and acoustic signals make them as attack targets during war time, which also challenge the performances of onboard acoustic instruments.
Up to now, many numerical methods are developed and then utilized in addressing the vibrational and acoustic characteristics of ship structures, which can be found in numerous literatures. Among them are finite element method (FEM), boundary element method (BEM), statistical energy method (SEA), and mesh free methods, etc. Through intense academic efforts from engineers and scientists, and also due to commercial operations from software developer, most them are coded into commercial software, and comprehensively influent the way we design our products almost covering all the engineering fields such as civil engineering, ship and ocean engineering, chemical engineering, and etc. To some extent, we must confess that we, not only engineers, scientists, but product managers, or even government officers, have underestimate the power of novel numerical methods and how much they forge the manufacturing process in modern industries.
As for finite element method, it is one of the most successful numerical methods in high fidelity modeling of the dynamic behaviors of complex structures. To the best of our knowledge, SAP is the first commercial software. Soon after, other software like ADINA, ANSYS, ABAQUS, NASTRAN, and DYNTRAN have been developed and scattered in worldwide universities and industries.
However, like any other numerical methods, FEM has many inherent drawbacks due to the way it discretizes the structures. For instance, to address the vibrational responses in high frequencies, the mesh size must be as tiny as 1/6, or less, of the structural waves so that it can accurately reproduce the dynamics of the structures. However, such a meshing strategy is not always successful since too much finer meshes need not only excessive computational costs, but also lead to unexpected numerical uncertainties.
As for ship structures, the vibration of fluidloaded plates or shells composes as a very important part in the studies of many engineering structures [1, 2, 3]. One of the major reasons lies in the fact that the dynamics of these structures depends on the structures and the fluid simultaneously. The vibrating structures can induce pressure disturbances in their surrounding fluid, and, in return, the resonance frequencies and vibrational responses of the structures can be altered [4, 5].
Recently, dynamic stiffness method (DSM) has won great interests and received intense studies [6, 7, 8, 9, 10] from research and design engineers because it can overcome the above issues without too much geometrical discretization requirements. Various DSM elements have been developed for transverse or inplane vibrations of plates. In the beginning, more research works were mainly focused on transverse vibrations since bending modes are easily excited, especially in low frequencies. Dozens of investigator [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] made comprehensive contributions on DSM that only accounts for transverse vibrations of a plate with two opposite edges simply supported. Later, Bercin and Langley [8, 9] proposed a DSM that incorporates both inplane and bending vibrations. It is reasonably expected that all these works are only applicable to few specified cases due to oversimplified modeling assumptions. To address the vibrations of more practical engineering structures, Casimir et al. [7] developed DSM elements for a plate with completely free boundary conditions, in which Gorman’s superposition method was employed to obtain the exact transverse displacements. Banerjee and his colleagues [10, 11, 12] proposed the dynamic stiffness matrix for a rectangular plate with arbitrary boundary conditions. Similarly to DSM for bending plates with arbitrary boundary conditions, the dynamic stiffness matrix for inplane vibrations of plates is developed by Ghorbel et al. [15, 16], NefovskaDanilovic and Petronijevic [17, 18] in which all the four edges can be prescribed with any arbitrary conditions by adopting Gorman’s superposition method.
Since the year 2016, Yin and his associates [19, 20, 21] have conducted comprehensive studies on developing dynamic stiffness method and its application to the dynamics of ship structures. Li et al. [19] proposed a dynamic stiffness formulation accounting for both inplane and bending vibrations of plates with two opposite edges simply supported. This method was then employed for modeling vibration transmission with builtup plate structures [22] and a ship cabin with complex hulls. To consider the dynamics of stiffened plates, Yin et al. [21] extended Li’s formulations and developed a dynamic stiffness method that considers torsion, bending, and extension vibrations in beams with eccentric crosssections.
The main objective of this work is to formulate the vibration analysis of ship structures based on dynamic stiffness method that accounts for both inplane and bending vibrations within plate itself, all possible motions in stiffened beams, fluidloading, respectively. The present paper is organized as follows. In Section 2, this dynamic stiffness method is briefly summarized, which present the development of the three types of models. In Section 3, our proposed method is demonstrated by investigating the characteristics of representative plate structures.
2. Development of dynamic stiffness formulations
2.1 Model description
Figure 1 shows multiple rectangular plates in global coordinates OXYZ, which are rigidly joined along their common edges. Each plate has dimension of
2.2 Development of plate element
Consider a vibrating flat plate in contact with acoustic fluid on its lower side, which is made of isotropic material with Young’s modulus E, bulk density
The extension rigidity B and flexural rigidity D can be found in Ref. [19].
According to Bercin and Langley [9], the displacements for the plate, which is simply supported along its two opposite edges, can be expressed as N truncation terms,
And,
where
where
Accordingly, the transverse shear force
Based on Eqs. (3) and (6), for any
Hence, the relationship between generalized displacements
2.3 Development of beam element
As shown in Figure 2, a beam with an eccentric cross section is located with geometric center O and the shear center G. Based on classical beam theory, the governing equations for the forced vibrations at line
where
Since the beam is attached to one edge of the plate, its motions are in the similar forms as that expressed in Eq. (3) and can be readily written as,
Substituting Eq. (10) into Eq. (9), and utilizing the orthogonality relationship of the modes, the vibration motions at the
Without complex derivation procedure, Eq. (11) can be rewritten in a more compact matrix form,
where the dynamic stiffness matrix has the following expressions:
2.4 Development of fluidloaded element: acoustic pressure
The acoustic pressure satisfies the Helmholtz equation,
where
where
where
It is noted that we have the expression
2.5 Dynamic responses of builtup plate structures
The dynamic stiffness matrices for the plate and the beam (in Sections 2.2 and 2.3) are expressed in local coordinates, which can be termed as local dynamic stiffness matrices. With reference to the conventional finite element technique, the dynamic stiffness matrix for each plate element and each beam element can be readily assembled into overall global dynamic stiffness matrix. Hence, the dynamic responses of a builtup structure composed of plates and beams can be solved through novel numerical methods.
3. Numerical results and discussion
Without loss of generality, we only focus on the vibration transmission in a builtup plate structure that is reinforced by stiffeners or plates. Numerical results for the dynamics of plates with beam stiffeners based on our method can found in [21].
3.1 Transmission modes within a plate stiffened by stiffeners
To demonstrate our method in addressing the vibration transmission within complex builtup structures, a horizontal plate reinforced by a vertical plate, i.e., plate 2 is employed in this subsection. The detailed parameters of the plates are listed in Table 1. The two opposite long edges of plate 1 is simply supported. One of the free end of the plate, namely, left edge, is subjected to uniformly distributed vertical forces of 1 N/m.









Plate 1  200  7800  0.3  0.01  6.0  1.0  0.008 
Plate 2  200  7800  0.3  0.01  0.5  1.0  0.008 
Yin et al. [22] identify that there are three representative transmission modes in a stiffened plate. As the plate structures get more complex, similar phenomena can be also found, in which a plate is stiffened by 9 identical plates. When the left side of the plate is enforced with transverse force, three representative transmission modes can be clearly identified. In Figure 3(a), only the left local portion of the plates is excited that implies bending waves cannot propagate effectively forward due the presence of the stiffening plates. However, in some frequency regimes as shown in Figure 3(b) and (c), bending waves can pass the stiffening members freely. As frequency increases, the stiffening members act more like a barrier that prevent structural waves propagate.
From Figure 3(a)–(d), we can convince that the vibration transmission modes do exist in even more complex plate structures. In addition, we suggest to explore the underlying mechanisms, if any, between these transmission modes and the wellknown pass band and stop band since vibration transmission is probably one of the most important characteristics in complex plate structures, e.g., ship structures, etc.
3.2 Vibrations of a ship hull in contact with water
Figure 4 shows a ship hull that is reinforced by eight beams with dimension
Figure 5 shows the curves for the vertical displacement obtained by FEM and DSM, respectively. The truncation term N is set to 6 in DSM and the mesh size in the FEM is
4. Conclusion
A DSM is proposed to address the dynamics of ship structures, which include three types of elements. First, a DSM formulation for both inplane and bending vibrations in flat rectangular plates is developed. Then, a DSM for stiffening beams is addressed, which accounts for all possible vibrations in plates and beams, i.e., bending, torsion, and extension motions. Finally, a DS plate element with fluid loadings included is formulated. The numerical results for the vibrations for a ship hull based on the proposed DSM have excellent agreement with those results obtained from FEM, which demonstrate its potential in addressing the dynamics of ship structures. In addition, vibration transmission modes of a stiffened plate are also addressed using this method.
Acknowledgments
This work was partially supported by HighTech Ship Fund from the Ministry of Industry and Information Technology (MIIT): Deepwater Semisubmersible Support Platform (No.: 2016 [546]), High Quality Brand Ship Board Machinery (No.: 2016 [547]). The authors would like to thank Provincial Youth Fund (No.: BK20170217) from Jiangsu Science Foundation.
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