Justify Assumption of Exponential Response to Free Vibration Problem

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• Trying2Learn
In summary, Euler's equation is a mathematical equation that governs the motion of an oscillating system. It can be solved for the oscillatory quantity ##\lambda## in a number of ways, depending on the type of damping present. If underdamping is present, the solution is of the form ##x = Ae^{at}\cos{(bt + \phi)}##.
Trying2Learn
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

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?

Trying2Learn said:
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.

vanhees71, hutchphd and 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!

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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!

Trying2Learn said:
(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.

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.

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

berkeman

1. What is an exponential response to free vibration problem?

An exponential response to free vibration problem is a mathematical model used to describe the behavior of a vibrating system over time. It assumes that the system's response to a disturbance follows an exponential decay or growth pattern.

2. Why is the assumption of exponential response used in free vibration problems?

The assumption of exponential response is used because it simplifies the mathematical analysis of the problem. It also provides a good approximation for many physical systems, such as mass-spring-damper systems.

3. What factors influence the validity of the exponential response assumption?

The validity of the exponential response assumption depends on the damping ratio, natural frequency, and initial conditions of the system. It is most accurate for systems with low damping and high natural frequency.

4. How is the exponential response assumption justified in practice?

The exponential response assumption is often justified through experimental data or by comparing the results to other analytical methods. It can also be verified by solving the differential equations of motion for the system.

5. Are there any limitations to using the exponential response assumption in free vibration problems?

While the exponential response assumption is a useful tool for analyzing free vibration problems, it may not accurately represent the behavior of highly damped or nonlinear systems. In these cases, other methods may be more appropriate.

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