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Dynamic response of Kirchhoff’s plates to transient hydrodynamic impact loads

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Abstract

The dynamic analysis of the elastic response of marine structures to intense, localized, transient, hydrodynamic impact loads is important in the design of marine vessels. A dry and wet dynamic analysis of vibration of a square Kirchhoff’ s plate is presented. The dry and wet natural frequencies and modeshapes of the plate have been evaluated by Galerkin’s method. The transient loads are of two kinds: a uniform stretching load and a hydrodynamic impact load (at different deadrise angles). The stretching load sets the plate into small-amplitude high-frequency vibrations. The computationally efficient normal mode analysis has been used to evaluate the dynamic deflections. Three different boundary conditions of the plate have been used. The response of the plate to these loads has been investigated for a wide range of impact speeds and material properties. The response characteristics have been established, namely (a) maximum deflection, (b) maximum dynamic overshoot relative to the static deflection, (c) quasi-static and dynamic zones of the response over the range of time-scales. The dependency of these characteristics on parameters like the deadrise angles, damping ratios, boundary conditions and forcing configurations have also been studied, to aid the designer. A separate study of the modal participation factor is used to establish modal truncation limits for the analysis, with subsequent increases in the computational efficiency.

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Abbreviations

L :

Length of the plate

B :

Width of the plate

c :

Damping

\(D = \frac{{E{h^3}}}{{12\left( {1 - {v^2}} \right)}}\) :

Flexural rigidity of the isotropic plate. Modulus of Elasticity

E :

Modulus of Elasticity

Y:

Y-axis (⊥r to the force sweep)

F(x,y,t):

Force

Fradiation(x,y,t):

Radiation Force

h :

Plate thickness

H jl :

Static amplitude of [ϕj(x)ϕl(y)]

A k jl :

Dyn. amplitude of [ϕj(x)ϕl(y)]

m :

Mass per unit surface area

q k (x,y) :

kth principal coordinate

t :

Time variable

T sp :

Splash time (wetting time)

V:

Vertical impact velocity

ψ(x,y,z,t):

Fluid velocity potential

Φk(x,y):

kth plate modeshape

G(x,y,z):

Green’s function

σ:

Source strength

r:

Distance between the field point and the source point

g:

Acceleration due to gravity

ρ:

Density of water

x :

Space variable along L

y :

Space variable along B

z(x,y,t) :

Dynamic deflection

z st (x,y,t) :

Static deflection

X:

X-axis (along forcing)

Z:

Z axis. (Normal to the plate)

Fimpact(x,y,t):

Impact Force

β :

Deadrise angle

γ :

Beam wave number

ζ :

Damping ratio

υ :

Poisson’s ratio

τ :

Non-D splash time

ω k :

kth Natural frequency

ϕj(x):

jth 2-D modeshape along L

ϕl(y):

lth 2-D modeshape along B

Φk(x,y):

kth 3-D modeshape

ψ*(x,y,z,t):

Modal velocity potential

qk(t):

kth principal coordinate

ξ,η,ς:

Source point coordinates

dS:

Control surface

Gjl(x,y):

Galerkin’s pre-multiplier

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Datta, N., Troesch, A.W. Dynamic response of Kirchhoff’s plates to transient hydrodynamic impact loads. Mar. Syst. Ocean Technol. 7, 79–94 (2012). https://doi.org/10.1007/BF03449302

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