Dynamic Stiffness Method for Vibrations of Ship Structures

*Xuewen Yin, Kuikui Zhong, Zitian Wei and Wenwei Wu*

### **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 in-plane 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 built-up 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, beam-stiffened, 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.

fluid-loading, 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.

**Figure 1** shows multiple rectangular plates in global coordinates OXYZ, which are rigidly joined along their common edges. Each plate has dimension of *Lx* � *Ly* and thickness of *h*. Its two opposite edges marked by the symbol 'S-S' denote simply supported boundary conditions while the other two edges are arbitrary. In addition, each plate is reinforced by uniform eccentric beams, and in contact with acoustic

Consider a vibrating flat plate in contact with acoustic fluid on its lower side,

*∂*2 *v ∂x∂y* þ *mω*<sup>2</sup> *<sup>B</sup> <sup>u</sup>* <sup>¼</sup> <sup>0</sup>

*∂*2 *u ∂x∂y* þ *mω*<sup>2</sup> *<sup>B</sup> <sup>v</sup>* <sup>¼</sup> <sup>0</sup>

(1)

*<sup>D</sup>*∇<sup>4</sup>*<sup>w</sup>* � *<sup>m</sup>ω*<sup>2</sup>*<sup>w</sup>* ¼ �*pa*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, 0

which is made of isotropic material with Young's modulus E, bulk density *ρ*, Poisson's ratio *μ*, and damping ratio *η* . Its governing equations for both in-plane and

> *∂*2 *u <sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

> *∂*2 *v <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

**2. Development of dynamic stiffness formulations**

*Dynamic Stiffness Method for Vibrations of Ship Structures*

*DOI: http://dx.doi.org/10.5772/intechopen.91990*

**2.1 Model description**

fluid on its one side.

**Figure 1.**

**211**

*A built-up plate structure with beam stiffeners.*

**2.2 Development of plate element**

bending vibrations can be written as,

*∂*2 *u <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>1</sup>

8 >>>>>><

>>>>>>:

*∂*2 *v <sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>1</sup>

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 fluid-loaded plates or shells composes as a very important part in the studies of many engineering structures [1–3]. 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 [6–10] 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 in-plane 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 [6–15] 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 in-plane 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 [10–12] 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 in-plane vibrations of plates is developed by Ghorbel et al. [15, 16], Nefovska-Danilovic 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 [19–21] 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 in-plane and bending vibrations of plates with two opposite edges simply supported. This method was then employed for modeling vibration transmission with built-up 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 cross-sections.

The main objective of this work is to formulate the vibration analysis of ship structures based on dynamic stiffness method that accounts for both in-plane and bending vibrations within plate itself, all possible motions in stiffened beams,

fluid-loading, 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.
