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Here https://en.wikipedia.org/wiki/Harmonic_oscillator we have an equation for displacement of a mass on a string as a function of time.

${\displaystyle x(t)=A\cos \left({\sqrt {\frac {k}{m}}}t\right)}$

In the same wikipedia article it is mentioned a few times that a dampened spring-mass "system" can exponentially decay.

How is the above equation augmented to account for dampening?

It is simply by multiplying by $e^{-\lambda t}$?

If yes, how do we interpret/define our $\lambda$?

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In cases like this it is always beneficial to plot the things we are talking about. enter image description here The red curve is just the cosine wave and the green one is a cosine wave multiplied by an exponentially decaying envelope which is shown in blue.

Since the time between two troughs is the time period $T$, then the ratio of displacements at two instants of time separated by a time $T$ will be given by $e^{-\lambda T}$.

Physically what this the green curve is telling us is that the periodicity is unchanged however the amplitude is decaying over time. The oscillator isn’t going as far as it initially did. And this is what happens during damping. We would want to capture the rate of this amplitude decay in some neat mathematical form, thus the $e^{-\lambda T}$. This $\lambda$ is exactly the rate of decay of the amplitude.

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Start with the equations of motion

$$ m \ddot{x} = -k x - d \dot{x} \tag{1}$$

where $m$ is mass, $k$ is stiffness (force/distance) and $d$ is damping (force/speed).

Parametrize the problem with $k = m\, \omega_n^2$ and $d =2 \zeta m\, \omega_n$ where instead of specifying $m$, $k$ and $d$, you specify a system with $\omega_n$ undamped natural frequency and $\zeta$ damping ratio.

The equations of motion are

$$ \ddot{x} = -\omega_n^2\, x - 2 \zeta \omega_n\, \dot{x} \tag{2}$$

with the well known solution of the type

$$ x= X \exp\left(-\lambda t\right) \sin \left( \omega t \right)\tag{3}$$

with $\lambda = \zeta \omega_n$ and $\omega = \omega_n \sqrt{1-\zeta^2}$.

Expanded this is

$$ x= X \exp\left(-\tfrac{d}{2 m} t\right) \sin \left( t\sqrt{\tfrac{k}{m}-\tfrac{d^2}{4 m^2}} \right)\tag{4}$$

provided that $ d \leq 2 \sqrt{k m}$.

The solution for higher damping is

$$ x= X \exp\left(-\tfrac{d}{2 m} t\right) \sinh \left( t\sqrt{\tfrac{d^2}{4 m^2}-\tfrac{k}{m}} \right)\tag{6}$$

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For the equation of the form: $$ m\frac{d^{2}x}{dt^{2}} + b\frac{dx}{dt} + kx = 0 $$ we get, $$ \lambda = \frac{b}{2m},\ w^2 = \frac{k}{m} $$ Now, you can write your equations as: $$ x =A\sin(wt+\phi),\ A = A_0\ e^{-\frac{bt}{2m}} $$ This factor is basically introduced when there is a force which acts against the motion and whose magnitude is proportional to the instantaneous velocity of the body undergoing SHM. This can also be seen in the equation, if you look closely the force applied by the damping force is the extra factor and proportional to the velocity with the proportionality constant of b, ie, $F_{damping} = -bv$ . An example of this type of a damping force would be air drag.

Hope this helps!

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