let's assume we have a cantilever beam of length L, which is clamped at left end (x=0) and is loaded with a time-varying force $F(t)$ at the free end (x=L). Further, the bending stiffness is $EI=const$, the cross-section $A=const$ and the material density is $\varrho$. The spatial and time depending deflection of the beam in transverse direction is $w(x,t)$. From the continuum mechanics, we know that the governing equation for the transverse vibrations of a dynamic beam of first order (=Euler-Bernoulli-Theory) is
$EI\frac{\partial^4 w(x,t)}{\partial x^4}+\varrho A \frac{\partial^2 w(x,t)}{\partial t^2}=p(x,t)$
which is a inhomogenous hyperbolic PDE with constant coefficients. Finally, the "load function" $q(x,t)$ on the right hand side can be considered as $q(x,t)=\delta (x-L)F(t)$
In the literature [1] the author solved similar problems using a "mixed" Laplace- and Fouriertransformation. In particular, the author uses the Laplace transformation for the time variable $t$ and the Fourier transformation for the spatial variable $x$. Please allow me to evaluate this on the stated problem above according to my understanding of [1]:
Step 1: Laplace-Transformation of the time variable:
$E I \frac{\partial^4 \bar{w}(x, s)}{\partial x^4}+s^2 \rho A \frac{\partial^2 \bar{w}(x, s)}{\partial t^2}=\bar{q}(x, s)$
Step 2: Fourier-Transformation of the spatial variable:
$E I a_n^4 W(n, s)+s^2\rho A W(n, s)=Q(n, s)$
whereby, according to the literature [1], $a_n=\frac{\pi n}{L}$
Step 3: Solution in frequency-spatial-domain
$W(n, s)=\frac{Q_n(n, s)}{\left[s^2 \rho A+E I a_n^4\right]}$
Without further thinking on the inverse transformation or the transformations of the "load function", I don't understand in what step (Transformation) the boundary conditions of the beam are considered, such as: deflection and slope at the clamped end = zero. For somehow, the author in [1] does not showing this or my mathematical comprehension is insufficient.
I apologize my English, since I am not native English speaking and further I want to apologize my mathematical skills, since I am just an engineer.
Literature:
[1]: Wave Motions in Elastic Solids - Karl F. Graff
Kind regards,
TheProtagonist
Your Laplace transform is wrong. The equation should read:
$EI\frac{\partial^4 \hat w}{\partial x^4}+\rho A s^2\hat w(x,s)=\hat q(x,s)$
Now in your Fourier transform since the domain is finite you have to take into account the boundary conditions.
Continue from here.
Further more you don't need Fourier transform since the force acts on right boundary, so the problem can be formulated:
$EI\frac{\partial ^4 w}{\partial x^4}+\rho A\frac{\partial^2 w}{\partial t^2}=0$
With boundary conditions: $w(0,t)=0, \frac{\partial w}{\partial x}|_0=0$
and: $EI\frac{\partial ^3 w}{\partial x^3}|_L=F(t), \frac{\partial ^2 w}{\partial x^2}|_L=0$
Since the force is prescribed there and the moment on $x=L$ is 0.