Justify Assumption of Exponential Response to Free Vibration Problem

[Trying2Learn]

TL;DR

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?

 

[BvU]

For example, could one say that the derivatives of the exponential are exponential and this serves as a basis for all possible functions?

You hit the nail on the head (old dutch expression).

In dealing with differential equations like $x'' + \omega^2 x = 0$ it is sensible to try exponentials. See characteristic equation in calculus books.

 

[etotheipi]

You don't need to assume any response. Newton's second law gives you a homogenous second order differential equation, $$m\ddot{x} + b \dot{x} + kx = 0$$ You can solve the complementary equation for $\lambda$ for the three different types of damping (depending on whether the discriminant $b^2 - 4mk$ is less than, equal to, or greater than zero) and come up with the equation of motion.

If you get complex $\lambda = a \pm bi$, the complementary solution is of the form $x = Ae^{at}\cos{(bt + \phi)}$. This is the case with underdamping!

 

[rude man]

OK, you need to know Euler's equation, a priori in your words.
Then the equation given in the previous post is solved in a number of ways. The "classical" way is to assume exponential solutions. Unexplained, a priori!

If you're working towards an electrical degree you will be saved with the use of the Laplace transform. Essentially you just look up the relevand Laplace-transformed term(s) in a nice table to get the time response. SO much easier and it includes all initial conditions FOR FREE!

 

[berkeman]

(Forgive me for marking this "Advanced" I seek a more conceptual understanding

No. "A" means you have a graduate school background and want to discuss the subject at the graduate school / PhD level. I will change it to "I" for you.

 

[Motore]

Here is a simple but intuitive way of thinking about Euler's equation and oscillations (vibrations):


I suggest watching from the beginning (and also previous lessons).
It's a bit on the basic level, but perhaps you will gain better insights.

 

[Trying2Learn]

thank you all. I just logged back in, and saw there were additional comments.
Thank you.