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I know that for the position $x$ as a function of time in an underdamped system (such as a mass on a spring) you can use the function:

$$x(t)=Ae^{\gamma t}cos(\omega t-\phi),$$

where

$$ \begin{split} &- A \text{ is the initial amplitude} \\ &-\gamma \text{ is the decay rate} \\ &-\omega \text{ is the natural frequency} \\ &-\phi \text{ is the initial phase angle}. \end{split}$$

This makes perfect sense to me and I understand all the variables. However, I have had difficulty finding a similar function for overdamped systems with a proper explanation. I know what a graph of such system looks like, but I am also interested in what a function for overdamped systems looks like. I am hoping someone could provide that.

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    $\begingroup$ It’s the same function, just increase the damping rate such that you get less than one oscillation. $\endgroup$
    – Gilbert
    May 21, 2022 at 0:39

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For the overdamped case the solutions are just exponential decays. There is no harmonic term. If you use complex represenation, for the underdamped case you have some imaginary exponent which results in harmonic functions. For the overdamped case the exponets are real so there is no harmonic part. Just exponential decay. The solutions are shown in most textbooks treating damped oscillations. And online you can see it for example here: https://beltoforion.de/en/harmonic_oscillator/

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