- #1
Trying2Learn
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- TL;DR Summary
- Why assume exponential
Consider a free vibration problem of a mass in 1D: inertia, damping, spring, and no forcing function.
We begin by assuming an exponential response.
And then run through all the cases of under, over, critical damping, etc.
I am fully aware of Euler's formula that relates sine/cosine to complex exponential, etc.
But when an analyst first attemps a solution of such a system, there is no a priori knowedge about such issues -- just a posteriori knowledge
of what it leads to.
Can someone justify why we assume a trial exponential response to a free vibration problem?
The only answer I seem able to give is that: it works because Euler, complex, trig, etc.
(Forgive me for marking this "Advanced" I seek a more conceptual understandingFor example, could one say that the derivatives of the exponential are exponential and this serves as a basis for all possible functions?
We begin by assuming an exponential response.
And then run through all the cases of under, over, critical damping, etc.
I am fully aware of Euler's formula that relates sine/cosine to complex exponential, etc.
But when an analyst first attemps a solution of such a system, there is no a priori knowedge about such issues -- just a posteriori knowledge
of what it leads to.
Can someone justify why we assume a trial exponential response to a free vibration problem?
The only answer I seem able to give is that: it works because Euler, complex, trig, etc.
(Forgive me for marking this "Advanced" I seek a more conceptual understandingFor example, could one say that the derivatives of the exponential are exponential and this serves as a basis for all possible functions?