Simulation of Flexural Dynamics of a Slender Ship undergoing Heave and Pitch

Abstract

A simulation model for the flexural dynamics of a slender ship undergoing heave and pitch in regular waves is presented. The heave and pitch motions are found from the strip theory of Gerritsma and Beukelman [1] which is known to be experimentally validated. It is assumed that the ship structure is sufficiently stiff so that bending deformations do not significantly alter the wave field and hence the hydrodynamic loads determined from the strip theory may be used as inputs to the flexural model. The flexural dynamics is modeled by applying the hydrodynamic loads to a lumped-mass beam model and the equations of motion are formulated using the methods of Kane and Levinson [2]. The flexural model can capture large deformation behavior and is tested by demonstrating convergence to a known analytic solution. The model is easily implemented and may be used to evaluate time histories of bending moment responses to waves as well as to slamming events if the slamming load is known or estimated. Some typical results are presented.

Appendices

Appendix A : steady state heave and pitch computation from Gerritsma and Beukelman strip theory [1]

The hydrodynamic forces on a strip at position s are given by Eq. (1). Summation of forces and moments over the length of the ship gives the equations of motions for heave and pitch in the form

$$\begin{aligned} \begin{array}{l} \left( M_{0}+M_{A}\right) \overset{\cdot \cdot }{z}+B_{z}\overset{\cdot }{z} +C_{z}z+D_{z}\overset{\cdot \cdot }{\theta }+E_{z}\overset{\cdot }{\theta } +H_{z}\theta =\overline{F}e^{-i\omega _{e}t}, \\ \left( I_{G}+I_{A}\right) \overset{\cdot \cdot }{\theta }+B_{\theta }\overset{\cdot }{\theta }+C_{\theta }\theta +D_{\theta }\overset{\cdot \cdot }{z} +E_{\theta }\overset{\cdot }{z}+H_{\theta }z=\overline{M}e^{-i\omega _{e}t}, \end{array} \end{aligned}$$

(A.1)

where

$$\begin{aligned} \begin{array}{l} M_{0}=\text {mass of ship}, \\ M_{A}=\int _{L}m_{a}\left( s\right) ds\ =\text {heave added mass}, \\ B_{z}=\int _{L}\left[ c\left( s\right) -V\frac{dm_{a}}{ds}\right] ds, \\ C_{z}=\rho g\int _{L}b\left( s\right) ds, \\ D_{z}=-\int _{L}sm_{a}\left( s\right) ds, \\ E_{z}=\int _{L}\left[ -sc\left( s\right) +2Vm_{a}\left( s\right) +Vs\frac{ dm_{a}}{ds}\right] ds, \\ H_{z}=-\rho g\int _{L}sb\left( s\right) \ ds+VB_{z}, \\ \overline{F}=\eta _{a}e^{-k_{e}T^{*}}\int _{L},\left[ \rho gb\left( s\right) -\omega ^{2}m_{a}\left( s\right) +i\omega \left( V\frac{dm_{a}}{ds} -c\left( s\right) \right) \right] e^{ik_{e}s}\ ds, \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} I_{G}=\text {pitch moment of inertia of ship about axis through }G, \\ I_{A}=\int _{L}s^{2}m_{a}\left( s\right) ds\ =\text {pitch added moment of inertia}, \\ B_{\theta }=\int _{L}\left[ s^{2}c\left( s\right) -2Vsm_{a}\left( s\right) -Vs^{2}\frac{dm_{a}}{ds}\right] ds, \\ C_{\theta }=\int _{L}\left[ \rho gs^{2}b\left( s\right) -Vsc\left( s\right) +V^{2}s\frac{dm_{a}}{ds}\right] ds, \\ D_{\theta }=-\int _{L}sm_{a}\left( s\right) ds, \\ E_{\theta }=\int _{L}\left[ -sc\left( s\right) +Vs\frac{dm_{a}}{ds}\right] ds, \\ H_{\theta }=-\rho g\int _{L}sb\left( s\right) \ ds, \\ \overline{M}=\eta _{a}e^{-k_{e}T^{*}}\int _{L}\left[ -\rho gsb\left( s\right) +\omega ^{2}sm_{a}\left( s\right) -i\omega s\left( V\frac{dm_{a}}{ds }-c\left( s\right) \right) \right] e^{ik_{e}s}\ ds. \end{array} \end{aligned}$$

Substituting $z=\overline{z}e^{-i\omega _{e}t}\ ,\ \theta =\overline{\theta } e^{-i\omega _{e}t}$ into equation (A.1) gives

$$\begin{aligned} R\left( \begin{array}{l} \overline{z} \\ \overline{\theta } \end{array} \right) =\left( \begin{array}{l} \overline{F} \\ \overline{M} \end{array} \right) , \end{aligned}$$

(A.2)

where the entries of matrix R are

$$\begin{aligned} \begin{array}{l} R_{11}=-\omega _{e}^{2}\left( M_{0}+M_{A}\right) -i\omega _{e}B_{z}+C_{z}, \\ R_{12}=-\omega _{e}^{2}D_{z}-i\omega _{e}E_{z}+H_{z}, \\ R_{21}=-\omega _{e}^{2}D_{\theta }-i\omega _{e}E_{\theta }+H_{\theta }, \\ R_{22}=-\omega _{e}^{2}\left( I_{G}+I_{A}\right) -i\omega _{e}B_{\theta }+C_{\theta }. \end{array} \end{aligned}$$

Equation (A.2) is solved for $\overline{z}$, $\overline{\theta }$. The steady state solutions for heave and pitch are found as the real parts of $\overline{z}e^{-i\omega _{e}t},\ \overline{\theta }e^{-i\omega _{e}t}$ and are given by

$$\begin{aligned} \begin{array}{l} z\left( t\right) =\alpha _{z}\cos \left( \omega _{e}t\right) +\beta _{z}\left( t\right) \sin \left( \omega _{e}t\right) , \\ \theta \left( t\right) =\alpha _{\theta }\cos \left( \omega _{e}t\right) +\beta _{\theta }\left( t\right) \sin \left( \omega _{e}t\right) , \end{array} \end{aligned}$$

where $\alpha _{z}=\hbox {Re}\left( \overline{z}\right) ,\ \beta _{z}=\hbox {Im }\left( \overline{z}\right) ,\ \alpha _{\theta }=\hbox {Re}\left( \overline{ \theta }\right) ,\ \beta _{\theta }=\hbox {Im}\left( \overline{\theta } \right)$.

Appendix B : determination of rotational spring constants

A non-uniform beam of length $L_{0}$ is cantilevered at end $C_{0}$ and subjected to a vertical load F at its free end. The beam is divided into n segments $S_{1},\ldots ,S_{n}$ by points $C_{0},C_{1},\ldots ,C_{n}$ and the length of segment $S_{k}$ is $\ell _{k}$. The segments are connected by rotational springs $\sigma _{1},\ldots ,\sigma _{n}$ as in Fig. 2. The spring $\sigma _{1}$ is located at $C_{0}$ and resists the rotation $\phi _{1}$ of segment $S_{1}$ relative to the horizontal. spring $\sigma _{k}$ is located at $C_{k-1}$ and resists the rotation $\phi _{k}\$of segment $S_{k}$ relative to segment $S_{k-1},\ k=2,\ldots ,n$. The vertical deflection at $C_{k}$ is $d_{k}\ \left( k-1,\ldots ,n\right)$ and may be computed by a standard finite element analysis. It is easy to show that

$$\begin{aligned} d_{k}=\sum _{r=1}^{k}\ell _{r}\mu _{r}, \end{aligned}$$

(B.1)

where

$$\begin{aligned} \mu _{r}=\sum _{i=1}^{r}\phi _{i}. \end{aligned}$$

(B.2)

Equation (B.1) may be written in the form of a lower triangular matrix and solved for $\mu _{1},\ldots ,\mu _{n}$. The segment rotations are then found from (B.2) as $\phi _{1}=\mu _{1},\ \phi _{k}=\mu _{k}-\mu _{k-1}$ for $k=2,\ldots ,n$. The spring constants are found from the relations $\sigma _{k}\phi _{k}=$ bending moment at $C_{k-1}$ $\left( k=1,\ldots ,n\right)$ . This gives

$$\begin{aligned} \sigma _{1}=\frac{FL_{0}}{\phi _{1}}\ \ ;\ \ \sigma _{k}=\frac{F}{\phi _{k}} \left( L_{0}-\sum _{r=1}^{k-1}\ell _{r}\right) \end{aligned}$$