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1- Elastodynamic equation

Navier Stokes' Equation :

1-Conservation of mass 2- Momentum equations

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

\(\sum{F} = ma\) 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:

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

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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