I am studying the vibrating dynamics of beams. In its resting state, the beam is flat, on what I call the $x$ axis. The variable $w(x,t)$ measures the vertical deflection from that axis for any point along the axis at any time.
The differential equation governing the dynamics of the beam is the following
$$\left[a\partial^4_x + b\partial^2_t +c\partial_t\right]w(x,t) = 0$$
with $a,b,c \in \mathbb{R}$ and $(x,t) \in [0,L]x[0,\infty[$
Literature can be found that deals extensively with the solutions of this differential equation. However, I am stuck because of my boundary conditions. The complete set of boundary conditions would be
$\left\{ \begin{array}{ll} w(0,t) = Acos(\Omega t) & \partial^2_x w(L,t) = 0 \\ \partial_x w(0,t) = 0 & \partial^3_x w(L,t) = 0 \end{array}\right. $
where the first column means that the beam is driven harmonically up and down on its left end, and the second column means that its other end is free (basically absence of bending and shear forces).
I am stuck because the usual procedure would be to separate variables. Assuming $w(x,t) := X(x)T(t)$ one ends up with
$$\frac{\partial^4_x X}{X} = -\frac{b\left[\partial^2_t{T}\right] + c\left[\partial_t{T}\right]}{aT} \overset{!}{=} \lambda$$
Where lambda is some constant to evaluate from the BCs. From my first boundary condition, I conclude that $T(t) = cos(\Omega t)$ and plugging it into the equation above, I am left with
$$\frac{\Omega^2b+\Omega c \tan(\Omega t)}{a} \overset{!}{=} \lambda$$
so I am toast since my left hand term is time-dependent.
Question: Do I need to investigate eigen function expansion as suggested here or is legitimate to make the substitution $w(x,t) = \text{Re}\{\tilde{w}(x,t)\}$ and change my first BC to $\tilde{w}(0,t) = Ae^{i\Omega t}$ which would then lead me to $\lambda/a = \Omega^2b -i\Omega c$, which indeed is constant, and carry on the usual way ?
Any feedback appreciated
