Alas, most oscillations are fleeting as the collective dance of energy between kinetic and potential gives way to the entropy gods. The culprit is usually a form of friction, which takes the ordered energy and transforms it into intractable random motion of atoms. Ultimately it ends up as thermal energy.
Lecture 3 | Damped Oscillations, Driven Oscillations & Resonance
Oscillating Systems
Lecture 3 | Damped Oscillations, Driven Oscillations & Resonance
Oscillating Systems
Pre-lecture Study Resources
Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.
Oscillations | Damped Simple Harmonic Oscillators
Oscillations in the real world seldom last without energy being put into maintaining their motion. This is due to frictional forces where the collective energy of the oscillator is lost to intractable forms, like thermal energy.
The amplitude of the oscillation decreases with time. If the force that causing the damping is linearly dependent on the velocity of the oscillator, the motion can be solved for using Calculus.
The solution to the equation of the amplitude as a function of time is exponentially decaying with time. In the above case, the amplitude of the oscillation A, changes with time. The sinusoidal part of the motion, the sine or cosine of omega ($\omega$) multiplied by time (t), with a possible phase shift $\phi_{0}$, modulates between positive and negative one - it provides the oscillatory motion in the mathematical model, but it needs to be scaled by the amplitude $A_{0}e^{-\beta t}$.
Finding the damping constant $\beta$ (equal to $\frac{1}{\tau}$ in BoxSand's notes) is often the goal, because that way you predict the amplitude at all times. The damping constant determines the extremity of the decay and may be considered analogous to finding the slope of a linear line. Once you know the slope of a line and one of it's points, you can predict the value of the function for all times. You can do this with an exponential function as well.
Key Equations and Infographics
Now, take a look at the pre-lecture reading and videos below.
