Calculating Transverse Wave Propagation in a Semi-Infinite Beam

[Trevorman]

1. A transversely directed transient force F(t) acts at the free end of a semi-infinite beam.
a) Show how displacement, velocity, acceleration and strain at an arbitrary position along the beam can be determined.
b) Calculate (MATLAB) the transversal acceleration (or an other quantity) at an arbitrary position. Assume suitable parameters and a force history.
c) Make the calculation for several positions thus illustrating the propagation of the wave.

Homework Equations

$\hat{v} = Ae^{i\beta x} + Be^{-i\beta x} +Ce^{-\beta x} + De^{\beta x}$
$\hat{T} = -E I \hat{v}^{\prime \prime \prime} = -E I \frac{\partial^3 \hat{v}}{\partial x^3}$
$\hat{T} = -\frac{1}{2}\hat{F}$

$v = \sum_n \hat{v} e^{-i \omega t}$

Where $v$ is the displacement
$T$ transverse force in the beam (given from free body diagram)
$E$ Youngs modulus
A,B,C,D is just constants

The Attempt at a Solution

What i know is that I can calculate the axial velocity and acceleration
Velocity
$\dot{v} = \sum_n - \hat{v} \omega e^{-i \omega t}$
Acceleration
$\dot{v} = \sum_n \hat{v} \omega^2 e^{-i \omega t}$

Also, since it is a semi-infinite beam, there will only be a wave going in one direction.
therefore the
$\hat{v} = A e^{-\beta x} + B e^{- i \beta x}$

I need to relate this to the force acting on the beam and do not know how to proceed...

 

[Trevorman]

The wave equation which is the simplest dynamic model for transverse motion of a beam is displayed below if it helps.

$v^{\prime \prime \prime \prime} + \frac{\rho A}{E I} \ddot{v} =0$

 

[Trevorman]

For those who have interest in this, I can give you the answer right away. Derivate the displacement with the boundary conditions, solve out all constants and you will get a very pretty expression that looks like this

$\hat{v}(x,\omega) = \frac{\hat{F}(\omega)}{4(EI)^{\frac{1}{4}}i \omega^{\frac{3}{2}}(\rho A)^{\frac{3}{4}}}\left[ e^{-i\left[\omega^2 \frac{\rho A}{EI} \right]^{\frac{1}{4}}x} -ie^{-\left[\omega^2 \frac{\rho A}{EI} \right]^{\frac{1}{4}}x} \right]$
Which could be expressed
$\hat{v}(x,\omega) = \hat{F}(\omega) \cdot \hat{H}(x,\omega)$

 

[Hugo Danielson]

Hej Trevor, did you ever manage with the MATLAB script?
Im doing a similar asssignment, and I am not getting the script to behave properly.