Fluid Structure Interaction:

Modelling the Structure:

1- Elastodynamic equation

Modelling the Fluid:

Navier Stokes' Equation :

1-Conservation of mass 2- Momentum equations

Modelling Mesh in common:

1- Different Coupling Methods could be used : ALE - arbitrary langrangian equations

Structural Dynamics:

Case of a continuous bending beam:

$\sum{F} = ma$ Looking at the transverse displacement of the string where $\sum{F}= b(x,t) - {d^2M \over dx^2} = b(x,t) - {d^2\over dx^2}(EI(x) {d^2u \over dx^2})$

Assuming an isotropic homogeneous material. $\sum{F} = b(x,t) - EI( {d^4u\over dx^4})$

Therefore the dynamic equilibrium equation becomes: $b(x,t) - EI( {d^4u\over dx^4}) = -\rho A u^{..}$

In the case of free vibration the equation becomes: $EI( {d^4u\over dx^4}) = \rho A u^{..}$

the general solution is found assuming an ansatz of $u(x,t) = \phi (x) \sin{wt}$

which is $\phi (x) = C_1 \cos{kx} + C_2 \sin{kx} + C_3 \cosh{kx}+ C_4 \sinh{kx}$ where $C_{i}$'s are constants of integration

for a continuous beam: assuming $u(x,t)=\phi (x) sin(wt)$

Under Free Vibration--

${EI d^4 \phi \over dx^4}- \rho A \omega ^2 \phi =0$

$phi'(x) = k(C1coskx+C2sinkx+C3coshkx+C4sinhkx)$

$phi''(x) = k^2(C1coskx+C2sinkx+C3coshkx+C4sinhkx)$

$phi'''(x) =k^3(C1coskx+C2sinkx+C3coshkx+C4sinhkx)$

Simply Supported Beam Under Travelling Harmonic Load:

Response to a periodic excitation:

Idealization of vortex-shedding as a sinusoidal load :

Considering an harmonic sinusoidal model, the root mean square (RMS) time-varying vortex shedding induced load acting at a particular location, x, along a member could be modelled as :

$F_{s}(x,t)= {\rho V^2C_{L}^~ D(x) sin[2 \pi \omega _{e}(x)t ] \over 2 }$

where: ρ= air mass density, taken as 1.29 [kg/m3];

V= the mean wind speed at location x, [m/s];

$C̃_{L}$= RMS lift (across-wind) force coefficient for the cross-sectional geometry a

x= coordinate describing length along the member, [m];

D(x)= the diameter of frontal width of a member at location x, [m];

ωe(x)= frequency at which vortex shedding occurs at location x, [Hz];

t= time, [s]. Source : http://www.wceng-fea.com/vortex_shedding.pdf

Static Aeroelasticity

The interaction of aerodynamic loading induced by steady ﬂow and the resulting elastic deformation of the lifting-surface structure:

Wall Mounted Model

Assume a beam is pivoted at a point O. Carrying out moment equilibrium at the pivot : $\sum{M}=0$

$M_{ac} + L(x_{0} − x_{ac})− W(x_{0} − x_{cg})−kθ = 0$ where L: is the aerostatic lifting force, M_{ac} is the aerostatic moment

and $x_{ac}$ is the center of aerostatic loading, while $x_{cg}$ is the center of gravity of the beam.

For linear aerodynamics, the lift for a rigid support is simply $L_{rigid} = q S C_{Lα} α_r$ whereas the lift for an elastic support is $L= q S C_{Lα} (α_{r} +θ)$ where $q = {1 \over 2} ρ_{\infty} U^2$ is the freestream dynamic pressure (${N \over m^2}$)

where $ρ_{\infty}$ is the freestream air density (${kg \over m^3}$) and U is the freestream air speed (${m \over s}$)

S is the planform area ($m^2$), and $C_{Lα}$ is the wing lift-curve slope

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