- #1
Trevorman
- 22
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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.
##\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
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...
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...
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