**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 *ρ*, Poisson's ratio *μ*, and damping ratio *η* . Its governing equations for both in-plane and bending vibrations can be written as,

$$\begin{cases} \frac{\partial^2 u}{\partial \mathbf{x}^2} + a\_1 \frac{\partial^2 u}{\partial y^2} + a\_2 \frac{\partial^2 v}{\partial \mathbf{x} \partial y} + \frac{m \alpha^2}{B} u = 0\\ \frac{\partial^2 v}{\partial y^2} + a\_1 \frac{\partial^2 v}{\partial \mathbf{x}^2} + a\_2 \frac{\partial^2 u}{\partial \mathbf{x} \partial y} + \frac{m \alpha^2}{B} v = 0\\ D\nabla^4 w - m \alpha^2 w = -p\_a(\mathbf{x}, y, \mathbf{0}) \end{cases} \tag{1}$$

**Figure 1.** *A built-up plate structure with beam stiffeners.*

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

*pa* is the induced acoustic pressure due to the bending vibration of the plate. u, v and w are the displacements in x-, y- and z-directions. *m* and *ω* are mass per unit area of the plate and circular frequency, respectively. The parameters *a*<sup>1</sup> and *a*<sup>2</sup> in Eq. (1) are defined as

$$a\_1 = \frac{1-\mu}{2}, a\_2 = \frac{1+\mu}{2} \tag{2}$$

*:*

Based on Eqs. (3) and (6), for any *n*th mode, the generalized displacement

Ð *Ly*

8

2 *Ly*

2 *Ly*

2 *Ly*

> 2 *Ly*

2 *Ly*

2 *Ly*

2 *Ly*

> 2 *Ly*

*u*ð Þ 0, *y* sin ð Þ *kny dy*

9

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=

, (7)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

8�1

9

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=

(8)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

8�1

*v*ð Þ 0, *y* cosð Þ *kny dy*

*w*ð Þ 0, *y* sin ð Þ *kny dy*

*θ*ð Þ 0, *y* sin ð Þ *kny dy*

*u L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy*

*v L*ð Þ *<sup>x</sup>*, *y* cosð Þ *kny dy*

*w L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy*

*θ*ð Þ *Lx*, *y* sin ð Þ *kny dy*

*N*ð Þ 0, *y* sin ð Þ *kny dy*

*T*ð Þ 0, *y* cosð Þ *kny dy*

*S*ð Þ 0, *y* sin ð Þ *kny dy*

*M*ð Þ 0, *y* sin ð Þ *kny dy*

*N L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy*

*T L*ð Þ *<sup>x</sup>*, *y* cosð Þ *kny dy*

*S L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy*

*M L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy*

0

Ð *Ly*

0

Ð *Ly*

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

0

Ð *Ly*

0

Ð *Ly*

0

Ð *Ly*

0

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

Ð *Ly*

0

Ð *Ly*

0

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

Ð *Ly*

2 *Ly*

2 *Ly*

> 2 *Ly*

2 *Ly*

> 2 *Ly*

> 2 *Ly*

> > 2 *Ly*

2 *Ly*

0

Ð *Ly*

0

Ð *Ly*

0

Ð *Ly*

0

� Ð *Ly*

� Ð *Ly*

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

� Ð *Ly*

� Ð *Ly*

0

Hence, the relationship between generalized displacements *qn* and generalized forces *Qn* at any *n*th mode can be developed after simple matrix algorithm, which is generally known as dynamic stiffness matrix *Kn*. Once the dynamic stiffness matrix

0

0

0

vector *qn* and force vector *Qn* are written as,

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

*Dynamic Stiffness Method for Vibrations of Ship Structures*

*un*j*x*¼<sup>0</sup>

9

>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>;

¼

*vn*j *x*¼0

8

>>>>>>>>>>>>>>>>>>>><

*w*j *x*¼0

*θn* � � *x*¼0

*un*j *x*¼*Lx*

*vn*j *x*¼*Lx wn*j*<sup>x</sup>*¼*Lx*

>>>>>>>>>>>>>>>>>>>>:

*θn* � � *x*¼*Lx*

*Nn* � � *x*¼0 *Tn* � � *x*¼0 *Sn* � � *x*¼0 *Mn* � � *x*¼0 9

>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>;

¼

8

>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>:

*Nn* � � *x*¼*Lx*

*Tn* � � *x*¼*Lx*

*Sn* � � *x*¼*Lx*

*Mn* � � *x*¼*Lx*

*qn* ¼

*Qn* ¼

**213**

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,

*u x*ð Þ¼ , *<sup>y</sup>* <sup>P</sup>*<sup>N</sup> n*¼1 *<sup>C</sup>*1*<sup>n</sup>λ*1*ne<sup>λ</sup>*1*nx* <sup>þ</sup> *<sup>C</sup>*2*<sup>n</sup>λ*2*ne<sup>λ</sup>*2*nx* <sup>þ</sup> *<sup>C</sup>*3*nkne<sup>λ</sup>*3*nx* <sup>þ</sup> *<sup>C</sup>*4*nkne<sup>λ</sup>*4*nx* � � sin ð Þ *kny v x*ð Þ¼ , *<sup>y</sup>* <sup>P</sup>*<sup>N</sup> n*¼1 *<sup>C</sup>*1*nkne<sup>λ</sup>*1*nx* <sup>þ</sup> *<sup>C</sup>*2*nkne<sup>λ</sup>*2*nx* <sup>þ</sup> *<sup>C</sup>*3*<sup>n</sup>λ*3*ne<sup>λ</sup>*3*nx* <sup>þ</sup> *<sup>C</sup>*4*<sup>n</sup>λ*4*ne<sup>λ</sup>*4*nx* � � cosð Þ *kny w x*ð Þ¼ , *<sup>y</sup>* <sup>P</sup>*<sup>N</sup> n*¼1 ð Þ cosð Þ *α*1*nx A*1*<sup>n</sup>* þ sin ð Þ *α*1*nx A*2*<sup>n</sup>* þ cosh ð Þ *α*2*nx A*3*<sup>n</sup>* þ sinh ð Þ *α*2*nx A*4*<sup>n</sup>* sin ð Þ *kny if k*<sup>2</sup> ≥ *kn* 2 , 8 >>>>>>>>>< >>>>>>>>>: (3)

And, *if k*<sup>2</sup> <*kn* 2 , the bending vibrations can be expanded as near-field disturbance,

$$w(\mathbf{x}, \mathbf{y}) = \sum\_{\mathbf{n}=1}^{N} \left( \cosh\left(a\_{\mathbf{n}\mathbf{r}}\mathbf{x}\right) A\_{\mathbf{1}\mathbf{n}} + \sinh\left(a\_{\mathbf{1}\mathbf{n}}\mathbf{x}\right) A\_{\mathbf{2}\mathbf{n}} + \cosh\left(a\_{\mathbf{2}\mathbf{n}}\mathbf{x}\right) A\_{\mathbf{3}\mathbf{n}} + \sinh\left(a\_{\mathbf{2}\mathbf{n}}\mathbf{x}\right) A\_{\mathbf{4}\mathbf{n}}\right) \tag{4}$$

where *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ρω*<sup>2</sup>*=<sup>D</sup>* <sup>p</sup> and *kn* <sup>¼</sup> *<sup>n</sup>π=Ly*. *Cmn*, *<sup>m</sup>* <sup>¼</sup> 1, 2, 3, 4 and *Amn*, *<sup>m</sup>* <sup>¼</sup> 1, 2, 3, 4 are the unknown constants. Wavenumbers for in-plane and out-of-plane waves take the following forms:

$$\begin{cases} \lambda\_{1n,2n} = \pm \sqrt{{k\_n}^2 - {k\_L}^2}, \lambda\_{3n,4n} = \pm \sqrt{{k\_n}^2 - {k\_T}^2} \\\ k^2 \ge k\_n^{-2}, a\_{1n} = \sqrt{{k\_n}^2 - {k\_n}^2}, a\_{2n} = \sqrt{{k\_n}^2 + {k\_n}^2} \\\ k^2 < k\_n^{-2}, a\_{1n} = \sqrt{{k\_n}^2 - {k\_n}^2}, a\_{2n} = \sqrt{{k\_n}^2 + {k\_n}^2} \end{cases} \tag{5}$$

where *kL* <sup>2</sup> <sup>¼</sup> *ρω*<sup>2</sup> <sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> ð Þ*=E*, *kT* <sup>2</sup> <sup>¼</sup> <sup>2</sup>*ρω*<sup>2</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> <sup>=</sup>E*.

Accordingly, the transverse shear force *Qx* perpendicular to xy plane, the bending moment *Mxx*, longitudinal force *Nxx*, and in-plane shear force *Nxy* along the plate junctions can be derived as follows,

$$\begin{cases} Q\_{\rm x} = -D \left( \frac{\partial^3 w}{\partial \mathbf{x}^3} + (2 - \mu) \frac{\partial^3 w}{\partial \mathbf{x} \partial \mathbf{y}^2} \right) \\\\ M\_{\rm xx} = -D \left( \frac{\partial^2 w}{\partial \mathbf{x}^2} + \mu \frac{\partial^2 w}{\partial \mathbf{y}^2} \right) \\\\ N\_{\rm xx} = -B \left( \frac{\partial u}{\partial \mathbf{x}} + \mu \frac{\partial v}{\partial \mathbf{y}} \right) \\\\ N\_{\rm xy} = -Ba\_1 \left( \frac{\partial u}{\partial \mathbf{y}} + \frac{\partial v}{\partial \mathbf{x}} \right) \end{cases} \tag{6}$$

*pa* is the induced acoustic pressure due to the bending vibration of the plate. u, v and w are the displacements in x-, y- and z-directions. *m* and *ω* are mass per unit area of the plate and circular frequency, respectively. The parameters *a*<sup>1</sup> and

<sup>2</sup> , *<sup>a</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*

*<sup>C</sup>*1*<sup>n</sup>λ*1*ne<sup>λ</sup>*1*nx* <sup>þ</sup> *<sup>C</sup>*2*<sup>n</sup>λ*2*ne<sup>λ</sup>*2*nx* <sup>þ</sup> *<sup>C</sup>*3*nkne<sup>λ</sup>*3*nx* <sup>þ</sup> *<sup>C</sup>*4*nkne<sup>λ</sup>*4*nx* � � sin ð Þ *kny*

*<sup>C</sup>*1*nkne<sup>λ</sup>*1*nx* <sup>þ</sup> *<sup>C</sup>*2*nkne<sup>λ</sup>*2*nx* <sup>þ</sup> *<sup>C</sup>*3*<sup>n</sup>λ*3*ne<sup>λ</sup>*3*nx* <sup>þ</sup> *<sup>C</sup>*4*<sup>n</sup>λ*4*ne<sup>λ</sup>*4*nx* � � cosð Þ *kny*

ð Þ cosð Þ *α*1*nx A*1*<sup>n</sup>* þ sin ð Þ *α*1*nx A*2*<sup>n</sup>* þ cosh ð Þ *α*2*nx A*3*<sup>n</sup>* þ sinh ð Þ *α*2*nx A*4*<sup>n</sup>* sin ð Þ *kny*

, the bending vibrations can be expanded as near-field disturbance,

2 ,

ð Þ cosh ð Þ *α*1*nx A*1*<sup>n</sup>* þ sinh ð Þ *α*1*nx A*2*<sup>n</sup>* þ cosh ð Þ *α*2*nx A*3*<sup>n</sup>* þ sinh ð Þ *α*2*nx A*4*<sup>n</sup>* sin ð Þ *kny*

*ρω*<sup>2</sup>*=<sup>D</sup>* <sup>p</sup> and *kn* <sup>¼</sup> *<sup>n</sup>π=Ly*. *Cmn*, *<sup>m</sup>* <sup>¼</sup> 1, 2, 3, 4 and *Amn*, *<sup>m</sup>* <sup>¼</sup> 1, 2, 3, 4

<sup>2</sup> <sup>p</sup> , *<sup>λ</sup>*3*n*,4*<sup>n</sup>* ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �

*∂*2 *w ∂y*<sup>2</sup>

<sup>2</sup> <sup>p</sup> , *<sup>α</sup>*2*<sup>n</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � *<sup>k</sup>*<sup>2</sup> <sup>p</sup> , *<sup>α</sup>*2*<sup>n</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*kn* <sup>2</sup> � *kT* <sup>2</sup> p

*kn* <sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> <sup>p</sup>

*∂*3 *w ∂x∂y*<sup>2</sup>

*<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *kn* <sup>2</sup> p

*if k*<sup>2</sup> ≥ *kn*

are the unknown constants. Wavenumbers for in-plane and out-of-plane waves

, *<sup>α</sup>*1*<sup>n</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>k</sup>*<sup>2</sup> � *kn*

, *<sup>α</sup>*1*<sup>n</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *kn*

<sup>2</sup> <sup>¼</sup> <sup>2</sup>*ρω*<sup>2</sup>ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup> <sup>=</sup>E*. Accordingly, the transverse shear force *Qx* perpendicular to xy plane, the bending moment *Mxx*, longitudinal force *Nxx*, and in-plane shear force *Nxy* along the

> *w <sup>∂</sup>x*<sup>3</sup> <sup>þ</sup> ð Þ <sup>2</sup> � *<sup>μ</sup>*

*w <sup>∂</sup>x*<sup>2</sup> <sup>þ</sup> *<sup>μ</sup>*

� �

*∂x* þ *μ ∂v ∂y*

*∂u ∂y* þ *∂v ∂x* � �

� �

*<sup>λ</sup>*1*n*,2*<sup>n</sup>* ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *kn* <sup>2</sup> � *kL*

*Qx* ¼ �*<sup>D</sup> <sup>∂</sup>*<sup>3</sup>

*Mxx* ¼ �*<sup>D</sup> <sup>∂</sup>*<sup>2</sup>

*Nxx* ¼ �*<sup>B</sup> <sup>∂</sup><sup>u</sup>*

*Nxy* ¼ �*Ba*<sup>1</sup>

*k*<sup>2</sup> ≥ *kn* 2

8 >>><

>>>:

plate junctions can be derived as follows,

*k*<sup>2</sup> < *kn* 2

8

>>>>>>>>>>>>><

>>>>>>>>>>>>>:

<sup>2</sup> <sup>¼</sup> *ρω*<sup>2</sup> <sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> ð Þ*=E*, *kT*

<sup>2</sup> (2)

*:*

(3)

(4)

(5)

(6)

*<sup>a</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> � *<sup>μ</sup>*

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

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,

*a*<sup>2</sup> in Eq. (1) are defined as

*u x*ð Þ¼ , *<sup>y</sup>* <sup>P</sup>*<sup>N</sup>*

*v x*ð Þ¼ , *<sup>y</sup>* <sup>P</sup>*<sup>N</sup>*

*n*¼1

And, *if k*<sup>2</sup> <*kn*

*n*¼1

where *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

take the following forms:

where *kL*

**212**

*w x*ð Þ¼ , *<sup>y</sup>* <sup>X</sup>*<sup>N</sup>*

*w x*ð Þ¼ , *<sup>y</sup>* <sup>P</sup>*<sup>N</sup>*

8

>>>>>>>>><

>>>>>>>>>:

*n*¼1

*n*¼1

2

Based on Eqs. (3) and (6), for any *n*th mode, the generalized displacement vector *qn* and force vector *Qn* are written as,

*qn* ¼ *un*j*x*¼<sup>0</sup> *vn*j *x*¼0 *w*j *x*¼0 *θn* � � *x*¼0 *un*j *x*¼*Lx vn*j *x*¼*Lx wn*j*<sup>x</sup>*¼*Lx θn* � � *x*¼*Lx* 8 >>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>; ¼ Ð *Ly* 0 2 *Ly u*ð Þ 0, *y* sin ð Þ *kny dy* Ð *Ly* 0 2 *Ly v*ð Þ 0, *y* cosð Þ *kny dy* Ð *Ly* 0 2 *Ly w*ð Þ 0, *y* sin ð Þ *kny dy* Ð *Ly* 0 2 *Ly θ*ð Þ 0, *y* sin ð Þ *kny dy* Ð *Ly* 0 2 *Ly u L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy* Ð *Ly* 0 2 *Ly v L*ð Þ *<sup>x</sup>*, *y* cosð Þ *kny dy* Ð *Ly* 0 2 *Ly w L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy* Ð *Ly* 0 2 *Ly θ*ð Þ *Lx*, *y* sin ð Þ *kny dy* 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>; 8�1 , (7) *Qn* ¼ *Nn* � � *x*¼0 *Tn* � � *x*¼0 *Sn* � � *x*¼0 *Mn* � � *x*¼0 *Nn* � � *x*¼*Lx Tn* � � *x*¼*Lx Sn* � � *x*¼*Lx Mn* � � *x*¼*Lx* 8 >>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>; ¼ Ð *Ly* 0 2 *Ly N*ð Þ 0, *y* sin ð Þ *kny dy* Ð *Ly* 0 2 *Ly T*ð Þ 0, *y* cosð Þ *kny dy* Ð *Ly* 0 2 *Ly S*ð Þ 0, *y* sin ð Þ *kny dy* Ð *Ly* 0 2 *Ly M*ð Þ 0, *y* sin ð Þ *kny dy* � Ð *Ly* 0 2 *Ly N L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy* � Ð *Ly* 0 2 *Ly T L*ð Þ *<sup>x</sup>*, *y* cosð Þ *kny dy* � Ð *Ly* 0 2 *Ly S L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy* � Ð *Ly* 0 2 *Ly M L*ð Þ *<sup>x</sup>*, *y* sin ð Þ *kny dy* 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>; 8�1 (8)

Hence, the relationship between generalized displacements *qn* and generalized forces *Qn* at any *n*th mode can be developed after simple matrix algorithm, which is generally known as dynamic stiffness matrix *Kn*. Once the dynamic stiffness matrix is obtained, the dynamic responses resulted from excitations can be readily achieved after solving linear equations like those in conventional finite element methods [19].

### **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 *G* � *G*<sup>0</sup> are expressed as,

$$\begin{cases} \begin{aligned} \frac{\partial^2}{\partial \mathbf{y}^2} \left( E\_r I\_x \frac{\partial^2 u\_r}{\partial \mathbf{y}^2} \right) - m\_r \alpha^2 u\_r + m\_r \alpha^2 \mathbf{z}\_G \phi\_r &= P\_r \\\\ E\_r \frac{\partial^2 v\_r}{\partial \mathbf{y}^2} + \rho\_r \alpha^2 v\_r &= N\_r \\\\ \frac{\partial^2}{\partial \mathbf{y}^2} \left( E\_r I\_x \frac{\partial^2 w\_r}{\partial \mathbf{y}^2} \right) - m\_r \alpha^2 w\_r - m\_r \alpha^2 \mathbf{x}\_G \phi\_r &= Q\_r \\\\ \text{eff}\_t \frac{\partial^2 \phi\_r}{\partial \mathbf{y}^2} + I\_0 \alpha^2 \phi\_r - m\_r \alpha^2 \mathbf{x}\_G w\_r + m\_r \alpha^2 \mathbf{z}\_G u\_r &= T\_r \end{aligned} \end{cases} \tag{9}$$

*ur yr* � � <sup>¼</sup> <sup>P</sup> *N n*¼1

8

*Dynamic Stiffness Method for Vibrations of Ship Structures*

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

>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>:

*ErIzk*<sup>4</sup>

8 >>>><

>>>>:

*ErIzk*<sup>4</sup>

compact matrix form,

the following form:

**215**

*Krn* ¼

*ErIxk*<sup>4</sup>

�*GItk*<sup>2</sup>

�*ErAk*<sup>2</sup>

<sup>0</sup> �*ErAk*<sup>2</sup>

derived,

*vr yr* � � <sup>¼</sup> <sup>P</sup> *N n*¼1

*wr yr*

*ϕ<sup>r</sup> yr* � � <sup>¼</sup> <sup>P</sup> *N n*¼1

*<sup>n</sup>* <sup>þ</sup> *mrω*<sup>2</sup> � �*vrn* <sup>¼</sup> *Nrn*

� � <sup>¼</sup> <sup>P</sup> *N n*¼1

the modes, the vibration motions at the *n*th mode for the beam can be readily

*<sup>n</sup>* � *mrω*<sup>2</sup> � �*urn* <sup>þ</sup> *mrω*<sup>2</sup>*zGϕrn* <sup>¼</sup> *Prn*

*<sup>n</sup>* � *mrω*<sup>2</sup> � �*wrn* � *mrω*<sup>2</sup>*xGϕrn* <sup>¼</sup> *Qrn*

where the dynamic stiffness matrix has the following expressions:

0 0 *ErIxk*<sup>4</sup>

**2.4 Development of fluid-loaded element: acoustic pressure**

The acoustic pressure satisfies the Helmholtz equation,

*∂*2 *∂y*<sup>2</sup> þ

*∂pa ∂z* � �� � � � *z*¼0

> � � �

*∂*2 *∂z*<sup>2</sup> � �*pa* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

where *k*<sup>0</sup> is the acoustic wavenumber. The boundary condition at the interface

where *ρ*<sup>0</sup> is the density of the acoustic fluid. Since the acoustic pressure *pa* has

<sup>¼</sup> *<sup>ρ</sup>*0*ω*<sup>2</sup>

*∂*2 *∂x*<sup>2</sup> þ

*pa*ð Þ¼ *x*, *y*, *z pa*

between the plate and the fluid is expressed as

*urn* sin *knyr* � �

*vrn* cos *knyr* � �

*wrn* sin *knyr* � � (10)

(11)

(13)

*ϕrn* sin *knyr* � �

Substituting Eq. (10) into Eq. (9), and utilizing the orthogonality relationship of

*<sup>n</sup>* <sup>þ</sup> *<sup>I</sup>*0*ω*<sup>2</sup> � �*ϕrn* � *mrω*<sup>2</sup>*xGwrn* <sup>þ</sup> *mrω*<sup>2</sup>*zGurn* <sup>¼</sup> *Trn*

*<sup>n</sup>* � *mrω*<sup>2</sup> 0 0 *mrω*<sup>2</sup>*zG*

*mrω*<sup>2</sup>*zG* <sup>0</sup> �*mrω*<sup>2</sup>*xG* �*GItk*<sup>2</sup>

*Frn* ¼ *Krn*ð Þ *ω qrn*, (12)

*<sup>n</sup>* � *mrω*<sup>2</sup> �*mrω*<sup>2</sup>*xG*

<sup>0</sup>*pa* ¼ 0, (14)

*w*, (15)

� exp �*j kxx* <sup>þ</sup> *kyy* <sup>þ</sup> *kzz* � � � � , (16)

*<sup>n</sup>* <sup>þ</sup> *<sup>I</sup>*0*ω*<sup>2</sup>

*<sup>n</sup>* <sup>þ</sup> *mrω*<sup>2</sup> <sup>0</sup>

Without complex derivation procedure, Eq. (11) can be rewritten in a more

where *ur*, *vr* and *wr* are the displacements in *xr*-, *yr*- and *zr*-directions, and *ϕ<sup>r</sup>* is the rotation about *yr* axis. *Pr*, *Nr*, *Qr* are the forces acting line *G* � *G*<sup>0</sup> in *xr*-, *yr*- and *zr*-directions, and *Tr* is the torsion moment about *yr* axis. *Ix* and *Iz* are the principle moments of the beam's cross-section about *xr*- and *zr*-axes. *Er* and *ρ<sup>r</sup>* are Young's modulus and density of the material. *mr* is mass per unit length of the beam, i.e., *ρrAr*, where *Ar* is the cross-sectional area. *G* and *I*<sup>0</sup> are shear modulus of the material, polar moment of mass inertia with respect to shear center, respectively, and *It* is cross-sectional factor in torsion.

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,

#### **Figure 2.**

*Schematic illustration of a beam: Geometric center* G*, shear center* O*; xG and zG are the offset between* G *and* O *in xr-, and zr-directions, respectively.*

*Dynamic Stiffness Method for Vibrations of Ship Structures DOI: http://dx.doi.org/10.5772/intechopen.91990*

is obtained, the dynamic responses resulted from excitations can be readily achieved after solving linear equations like those in conventional finite element

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

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 *G* � *G*<sup>0</sup> are expressed as,

� *mrω*<sup>2</sup>

� *mrω*<sup>2</sup>

*<sup>ϕ</sup><sup>r</sup>* � *mrω*<sup>2</sup>

*vr* ¼ *Nr*

where *ur*, *vr* and *wr* are the displacements in *xr*-, *yr*- and *zr*-directions, and *ϕ<sup>r</sup>* is the rotation about *yr* axis. *Pr*, *Nr*, *Qr* are the forces acting line *G* � *G*<sup>0</sup> in *xr*-, *yr*- and *zr*-directions, and *Tr* is the torsion moment about *yr* axis. *Ix* and *Iz* are the principle moments of the beam's cross-section about *xr*- and *zr*-axes. *Er* and *ρ<sup>r</sup>* are Young's modulus and density of the material. *mr* is mass per unit length of the beam, i.e., *ρrAr*, where *Ar* is the cross-sectional area. *G* and *I*<sup>0</sup> are shear modulus of the material, polar moment of mass inertia with respect to shear center, respectively,

Since the beam is attached to one edge of the plate, its motions are in the similar

*Schematic illustration of a beam: Geometric center* G*, shear center* O*; xG and zG are the offset between* G *and* O

*ur* <sup>þ</sup> *mrω*<sup>2</sup>

*wr* � *mrω*<sup>2</sup>

*xGwr* <sup>þ</sup> *mrω*<sup>2</sup>

*zGϕ<sup>r</sup>* ¼ *Pr*

*xGϕ<sup>r</sup>* ¼ *Qr*

*zGur* ¼ *Tr*

(9)

methods [19].

**2.3 Development of beam element**

8

>>>>>>>>>>>>><

>>>>>>>>>>>>>:

*∂*2 *<sup>∂</sup>y*<sup>2</sup> *ErIz*

*∂*2 *<sup>∂</sup>y*<sup>2</sup> *ErIx*

*GIt ∂*2 *ϕr <sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>I</sup>*0*ω*<sup>2</sup>

and *It* is cross-sectional factor in torsion.

**Figure 2.**

**214**

*in xr-, and zr-directions, respectively.*

*Er ∂*2 *vr <sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>ρ</sup>rω*<sup>2</sup>

*∂*2 *ur ∂y*<sup>2</sup> � �

*∂*2 *wr ∂y*<sup>2</sup> � �

forms as that expressed in Eq. (3) and can be readily written as,

$$\begin{cases} \begin{aligned} \boldsymbol{u}\_{r}(\boldsymbol{y}\_{r}) &= \sum\_{n=1}^{N} \boldsymbol{u}\_{m} \sin \left(k\_{n} \boldsymbol{y}\_{r}\right) \\\\ \boldsymbol{v}\_{r}(\boldsymbol{y}\_{r}) &= \sum\_{n=1}^{N} \boldsymbol{v}\_{m} \cos \left(k\_{n} \boldsymbol{y}\_{r}\right) \\\\ \boldsymbol{w}\_{r}(\boldsymbol{y}\_{r}) &= \sum\_{n=1}^{N} \boldsymbol{w}\_{m} \sin \left(k\_{n} \boldsymbol{y}\_{r}\right) \\\\ \boldsymbol{\phi}\_{r}(\boldsymbol{y}\_{r}) &= \sum\_{n=1}^{N} \boldsymbol{\phi}\_{m} \sin \left(k\_{n} \boldsymbol{y}\_{r}\right) \end{aligned} \end{cases} \tag{10}$$

Substituting Eq. (10) into Eq. (9), and utilizing the orthogonality relationship of the modes, the vibration motions at the *n*th mode for the beam can be readily derived,

$$\begin{cases} \left(E\_r I\_z k\_n^4 - m\_r \alpha^2\right) u\_m + m\_r \alpha^2 \mathbf{z}\_G \boldsymbol{\phi}\_m = P\_m \\ \left(-E\_r A k\_n^2 + m\_r \alpha^2\right) v\_m = N\_m \\ \left(E\_r I\_x k\_n^4 - m\_r \alpha^2\right) w\_m - m\_r \alpha^2 \mathbf{x}\_G \boldsymbol{\phi}\_m = \mathbf{Q}\_m \\ \left(-G I\_t k\_n^2 + I\_t \alpha^2\right) \boldsymbol{\phi}\_m - m\_r \alpha^2 \mathbf{x}\_G w\_m + m\_r \alpha^2 \mathbf{z}\_G \boldsymbol{u}\_m = T\_m \end{cases} \tag{11}$$

Without complex derivation procedure, Eq. (11) can be rewritten in a more compact matrix form,

$$
\overline{F}\_m = \overline{K}\_m(a)\overline{q}\_m,\tag{12}
$$

where the dynamic stiffness matrix has the following expressions:

$$
\overline{K}\_{rn} = \begin{bmatrix}
E\_r I\_z k\_n^4 - m\_r o^2 & 0 & 0 & m\_r o^2 \mathbf{z}\_G \\
0 & -E\_r A k\_n^2 + m\_r o^2 & 0 \\
0 & 0 & E\_r I\_x k\_n^4 - m\_r o^2 & -m\_r o^2 \mathbf{x}\_G \\
m\_r o^2 \mathbf{z}\_G & 0 & -m\_r o^2 \mathbf{x}\_G & -G I\_t k\_n^2 + I\_0 a^2
\end{bmatrix}.\tag{13}
$$

#### **2.4 Development of fluid-loaded element: acoustic pressure**

The acoustic pressure satisfies the Helmholtz equation,

$$
\left(\frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial \mathbf{y}^2} + \frac{\partial^2}{\partial \mathbf{z}^2}\right) p\_a + k\_0^2 p\_a = \mathbf{0},\tag{14}
$$

where *k*<sup>0</sup> is the acoustic wavenumber. The boundary condition at the interface between the plate and the fluid is expressed as

$$
\left.\left(\frac{\partial p\_a}{\partial \mathbf{z}}\right)\right|\_{\mathbf{z}=\mathbf{0}} = \rho\_0 a \rho^2 w,\tag{15}
$$

where *ρ*<sup>0</sup> is the density of the acoustic fluid. Since the acoustic pressure *pa* has the following form:

$$p\_a(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \left| p\_a \right| \exp \left[ -j \left( k\_x \mathbf{x} + k\_y \mathbf{y} + k\_z \mathbf{z} \right) \right], \tag{16}$$

where *pa* � � � � is the amplitude of the acoustic pressure, and *kx*, *ky*, and *kz* are wavenumbers for the acoustic waves. It is ready to obtain the expression for the acoustic pressure at the plate-fluid interface,

$$p\_a(\mathbf{x}, \mathbf{y}, \mathbf{0}) = \begin{cases} \frac{j\rho\_0 w^2 w}{\left(k\_0^2 - k\_b^2\right)^{\frac{1}{2}}}, & \text{if} \quad k\_b < k\_0 \text{ }, \\\frac{-\rho\_0 w^2 w}{\left(k\_b^2 - k\_0^2\right)^{\frac{1}{2}}}, & \text{if} \quad k\_b > k\_0 \text{ }. \end{cases} \tag{17}$$

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

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

**)** *μ η Lx***(m)** *Ly***(m)** *h***(m)**

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 well-known pass band and stop band since vibration transmission is probably one of the most important characteristics in complex plate structures,

*Representative vibrational transmission modes of a stiffened plate: (a) 270 Hz, (b) 345 Hz, (c) 395 Hz, and*

prevent structural waves propagate.

*No beam stiffeners are considered in case 1.*

**Table 1.**

**Figure 3.**

**217**

*(d) 445 Hz.*

*E* **(Gpa)** *ρ***(Kg/m<sup>3</sup>**

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

*Dynamic Stiffness Method for Vibrations of Ship Structures*

*Geometry and material parameters of the plates (case 1).*

e.g., ship structures, etc.

It is noted that we have the expression *k*<sup>2</sup> *<sup>b</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *<sup>y</sup>* , where *kb* is the wavenumbers for the structural waves propagating within the plates. For sake of brevity, the relationship between the acoustic pressure at the fluid–structure interface and the inertia terms due to the vibration of the plate, which is referred to as fluid-loading parameter, can be rewritten as,

$$\varepsilon\_{f} = p\_{a}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \not{p\_{\text{mo}}}\_{\left[\text{mo}^{2}w\right]} = \begin{cases} \frac{j\rho\_{0}}{m\left(k\_{0}^{2} - k\_{b}^{2}\right)^{1/2}}, & \text{if} \quad k\_{b} < k\_{0} \text{ }, \\\\ \frac{-\rho\_{0}}{m\left(k\_{b}^{2} - k\_{0}^{2}\right)^{1/2}}, & \text{if} \quad k\_{b} > k\_{0}. \end{cases} \tag{18}$$

#### **2.5 Dynamic responses of built-up 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 built-up 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 built-up 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 built-up 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.

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

*Dynamic Stiffness Method for Vibrations of Ship Structures DOI: http://dx.doi.org/10.5772/intechopen.91990*


**Table 1.**

where *pa* � � �

acoustic pressure at the plate-fluid interface,

*pa*ð Þ¼ *x*, *y*, 0

It is noted that we have the expression *k*<sup>2</sup>

fluid-loading parameter, can be rewritten as,

*ε <sup>f</sup>* ¼ *pa*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, 0

through novel numerical methods.

found in [21].

vertical forces of 1 N/m.

**216**

**3. Numerical results and discussion**

*=<sup>m</sup><sup>ω</sup>* ½ � <sup>2</sup>*<sup>w</sup>* ¼

**2.5 Dynamic responses of built-up plate structures**

� is the amplitude of the acoustic pressure, and *kx*, *ky*, and *kz* are

*=*2

> *=*2

*<sup>b</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

*jρ*0

�*ρ*<sup>0</sup>

, *if kb* <*k*<sup>0</sup> ,

, *if kb* > *k*<sup>0</sup> *:*

*<sup>y</sup>* , where *kb* is the

, *if kb* <*k*<sup>0</sup> ,

, *if kb* > *k*0*:*

*<sup>x</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

*=*2

> *=*2

(17)

(18)

wavenumbers for the acoustic waves. It is ready to obtain the expression for the

*jρ*0*ω*2*w*

�*ρ*0*ω*2*<sup>w</sup>*

wavenumbers for the structural waves propagating within the plates. For sake of brevity, the relationship between the acoustic pressure at the fluid–structure interface and the inertia terms due to the vibration of the plate, which is referred to as

> *m k*<sup>2</sup> <sup>0</sup> � *<sup>k</sup>*<sup>2</sup> *b* � �<sup>1</sup>

8 >>><

>>>:

*m k*<sup>2</sup> *<sup>b</sup>* � *<sup>k</sup>*<sup>2</sup> 0 � �<sup>1</sup>

The dynamic stiffness matrices for the plate and the beam (in Sections 2.2 and

Without loss of generality, we only focus on the vibration transmission in a built-up plate structure that is reinforced by stiffeners or plates. Numerical results for the dynamics of plates with beam stiffeners based on our method can

To demonstrate our method in addressing the vibration transmission within complex built-up 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

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

**3.1 Transmission modes within a plate stiffened by stiffeners**

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 built-up structure composed of plates and beams can be solved

*k*2 <sup>0</sup> � *<sup>k</sup>*<sup>2</sup> *b* � �<sup>1</sup>

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

8 >>>><

>>>>:

*k*2 *<sup>b</sup>* � *<sup>k</sup>*<sup>2</sup> 0 � �<sup>1</sup> *Geometry and material parameters of the plates (case 1).*

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 well-known 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.

**Figure 3.**

*Representative vibrational transmission modes of a stiffened plate: (a) 270 Hz, (b) 345 Hz, (c) 395 Hz, and (d) 445 Hz.*

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

This work was partially supported by High-Tech Ship Fund from the Ministry of Industry and Information Technology (MIIT): Deepwater Semi-submersible Support Platform (No.: 2016 [546]), High Quality Brand Ship Board Machinery (No.: 2016 [547]). The authors would like to thank Provincial Youth Fund (No.:

also addressed using this method.

BK20170217) from Jiangsu Science Foundation.

*Dynamic Stiffness Method for Vibrations of Ship Structures*

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

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

\*Address all correspondence to: x.w.yin@cssrc.com.cn

National Key Laboratory on Ship Vibration and Noise, China Ship Scientific

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Acknowledgements**

**Author details**

**219**

Research Center, Wuxi, China

provided the original work is properly cited.

**Figure 4.** *A ship hull reinforced with eight stiffeners.*

**Figure 5.** *Vertical displacement at the middle point in the bottom plate.*

#### **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 0*:*02*m* 0*:*02*m* along the junctions of their neural planes. The ship hull has the dimension of 6*m* 4*m* 2*:*4*m* and with thickness of 0.008 m. The bottom of the ship hull is in contact with water. About 1 N concentrated force is applied at the middle point in upper plate and the response gauge is set at middle point in the bottom plate.

**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 0*:*2*m* 0*:*2*m*. It is indicated that satisfactory agreement can be found between the results from DSM and those from FEM, which implies that our proposed method can provide excellent numerical results for ship structures.
