How can a decay factor be calculated

Damped vibration

Experiment: spring pendulum

A weight (orange box) hangs on a spring. If it is pulled down and then released, it begins to swing up and down.

Left: Vibration with friction
The vibration loses energy through friction, so the weight oscillates closer and closer to the rest position and finally stops vibrating.

Right: Vibration without friction
The weight oscillates evenly around the rest position.

In the chapter "Harmonic oscillation" we have dealt with oscillation without friction. Now it is the turn of the damped oscillation.

Loss of energy through friction

Physical systems always give off energy to their environment, e.g. through friction. It is therefore referred to as muffled. Leaving such a system to its own devices will ultimately lead to a standstill. Perpetua mobilia are therefore not possible (see the law of conservation of energy).

Application to the spring pendulum

A large part of the oscillation energy of the spring pendulum is converted into thermal energy when the spring is deformed. But air friction can also play a role (depending on the cross-section of the weight).

generalization

Most of the damped vibrations can be removed with the help of a Damping constant Describe \ (\ delta \) (also called the decay coefficient). This indicates how strongly the oscillation is damped.

If you look at how the damping constant is built into the oscillation equation, you can see that it does not change the sine function itself, but only the amplitude.

\ begin {aligned} s_ {harmonic} (t) & = \ underset {\ text {Amplitude}} {\ underbrace {\ hspace {1em} s_0 \ hspace {1em}}} \ cdot \ sin (\ omega t + \ phi_0) \ & \ s_ {damped} (t) & = \ underset {\ text {Amplitude}} {\ underbrace {s_0 \ cdot e ^ {- \ delta t}}} \ cdot \ sin (\ omega t + \ phi_0) \ \ end {aligned}

Amplitude function

The first part of the oscillation equation is also called the amplitude function: $$ \ hat {s} (t) = s_0 \ cdot e ^ {- \ delta t} $$

Left:
The amplitude function for different \ (\ delta \) (in gray).
It is easy to see how the amplitude decreases exponentially.

Special case \ (\ delta = 0 \):
The oscillation is undamped -> harmonious.

example 1:
\ (s_0 = 2 m \), \ (f = \ frac {1} {5} Hz \) and \ (\ phi_0 = 0 \) and \ (\ delta = 0.1 \)