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.

https://www.youtube.com/watch?v=HRcjtVa1LfM

## Pre-lecture Study Resources

Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.

### BoxSand Videos

## Required Videos

## Suggested Supplemental Videos

### Learning Objectives

## Summary

Summary

## Atomistic Goals

Students will be able to...

### BoxSand Introduction

## 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.

*credit MapleSoft

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.

### OpenStax Reading

OpenStax Section 16.7 | Damped Harmonic Motion

OpenStax Section 16.8 | Forced Oscillations and Resonance

## Additional Study Resources

Use the supplemental resources below to support your post-lecture study.

### YouTube Videos

This video gives a great lecture on damped oscillations

https://www.youtube.com/embed/rBokYpEze08?rel=0

Doc Schuster lecture on damped harmonic motion, using an analogy of a cars shock absorbers.

https://www.youtube.com/embed/H6wAgtvX_8w?rel=0

Michel van Biezen has, along with every other subject, a series of lectures about damping and simple harmonic motion(full list). This video is a solid example of an oscillatory motion problem with damping.

https://www.youtube.com/embed/USR7b_RuVss?rel=0

### Other Resources

This link will take you to the repository of other content related resources.

### Simulations

The simple pendulum is a special physical case that represents simple harmonic motion. We've used this simulation before, this time use the slider for friction to see the effect of damping.

For additional simulations on this subject, visit the simulations repository.

### Demos

### History

Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Physics Fun

Acoustic resonant modes of a levitating water droplet.

https://www.youtube.com/watch?v=JMmzCOGTbk0

Standing waves inside a barrel create a train horn.

https://www.youtube.com/watch?v=Rh9UwLb4Qh8

All you need to measure the speed of light is some Peeps, a microwave oven, and a little physics ingenuity.

https://www.youtube.com/watch?v=HwREvdUWSKE

Play the 2v4, why not what have you.

https://www.youtube.com/watch?v=qhqbSeG5DbA

How to break a wine glass with a megaphone.

### Other Resources

Hyperphysics' page on damped oscillations covers the different cases of damping.

The boundless section on damped oscillations is also followed by their section on driven oscillations

Boston University physics page which covers both damped and driven oscillations

Nice lecture slides which cover damped and driven oscillations.

### Other Resources

This link will take you to the repository of other content related resources .

## Resource Repository

This link will take you to the repository of other content related resources.

## Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

### Assumptions

Damping and driven oscillators rely on more than just the restoring force to dictate their movement. Driven oscillators are usually powered, increase in amplitude over time, and have energy entering the system while damped oscillators decrease in amplitude and are usually caused by some type of friction or drag force.

Although in our study of friction in 201 showed that the force from fiction was linear, damped oscillators in general do not decrease linearly in amplitude over time. They can in special cases, but it shouldn't be assumed that they will, it depends on how the dampening effect is generated and in this class will be tied up in the $\beta$ value in our equations to simplify studying the motion of damped oscillators.

### Checklist

Most damping problems involve using the damping equation: $$x(t) = x(t=0)e^{-\left( \beta t \right)}cos(\omega t)$$ Beta plays a central role in these problems, so it is important to know its value.

If you are told the original amplitude and the amplitude at a later time, you can quickly find beta. Just like you can use two values of a linear function to determine the value of the slope, knowing the amplitude at $t=0$ and at a later time tells you the value of beta - the only difference is in the algebra (you have to take a natural log).

For example, say you have a 500 g ball attached to a wall via a spring with spring constant $k=100$ N/m resting on a horizontal surface. You pull the ball back $0.8 cm$ and 20 full oscillations later its position is $x=0.5 cm$. In this case, we don't need to worry about the value of the cosine function: since we know this is $n$ *complete* cycles later, we know the function is at its max value - which implies $cos →1$.

To determine $\beta$, we do the following:

(1) Find the period: $$T= 2\pi \sqrt{\frac{m}{k}} $$ $$T = 0.444 s$$

(2) Use the period to find the time elapsed $$t = T*n_{cycles} = 0.444 * 20 = 8.886 s $$

(3) Solve for $\beta$:

$$x(t) = x(t=0)e^{-\left( \beta t \right)}$$

$$⇒ 0.5 \hspace{0.1cm} cm = 0.8 \hspace{0.1cm} cm \hspace{0.1cm} e^{-\left( \beta t \right)} $$

$$⇒ ln\left( \frac{0.5cm}{0.8cm}\right) = - \beta * 8.886 s $$

$$ ⇒ \beta = 5.3*10^{-3} s^{-1} $$

Now that you know $\beta$ you can find the value of $x_{max}(t)$ at any time $t$!

### Misconceptions & Mistakes

- Damping does not happen linearly. It takes a different amount of time for an amplitude to go from $x_{max}$ to $\frac{1}{2}x_{max}$ than from $\frac{1}{2}x_{max}$ to $0$.
- Damped motion is not necessarily oscillatory motion. For example, critically and over damped systems the amplitude goes from its maximum value to equilibrium position without ever going through the equilibrium location.

### Pro Tips

- You already know how to use two points from a linear line to find the slope of that line. Finding and using $\beta$ is analogous to this - in this analogy you can think of $\beta$ as the slope of the exponential. Knowing the value of $\beta$ allows you to predict the value of the function at a later time - just like knowing the slope of a linear line does. The only difference between finding the slope of a linear line and finding $\beta$ for an exponential is that you must use a natural log to get $\beta$ out of the exponential. After that, it's simple algebra to solve for $\beta$.
- Since many problems involving damping require using the equation $x(t) = x_{t=o}e^{-\frac{t}{\beta}}cos(\omega t)$

you should familiarize yourself with it. In which situations should you use sine instead of cosine? What does a decaying exponential look like, and how do you use the time constant to predict the value of the function at a later time? - A "damped" oscillation is the motion of an oscillating mass for which there is some frictional force (some force that always opposes the direction of motion); the only conceptual difference between a "damped" and "driven" oscillator is that, for a driven oscillator, the net external force is instead in the same direction of the motion of the object.

### Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

### Physical

One type of damping is drag force which come from changes in momentum. This could be caused by a ball colliding with air particles in the path of the balls motion.

### Mathematical

$x_{max}(t) = x_{max} (t=0) e^{-\frac{t}{\tau}}$

Position ($x$), Time ($t$), Damping Constant ($\tau$)

### Graphical

There are several different types of damping with respect to oscillatory motion: overdamped, underdamped, critically damped. The plot below describes the different types of damping and the type of motion that follows.

### Descriptive

Damping is ultamately cause by a force acting on system that decreases the acceleration of an object over time until the object stops moving or oscillating.

### Experimental

The following experiment demonstrates the physical motion for underdamping, crtical damping, and over damping of a simple harmonic oscillator. There are interactive icons on the left side of the video to view each type of damping.

## Practice

Use the practice problem sets below to strengthen your knowledge of this topic.

### Fundamental examples

(1) A spring with spring constant $k = 20.0 $ N/m hangs from the ceiling. A $600$ g ball is attached to the spring and allowed to come to rest. It is pulled down $8.0$ cm and released. What is the time constant if the ball’s amplitude has decreased to $4.0$ cm after $25$ oscillations?

(2) A spring with spring constant $k = 395$ N/m hangs from the ceiling. A $40$ g ball is attached to the spring and allowed to come to rest. It is then pulled down $6.0$ cm and released. If the damping constant is $\beta = \frac{1}{30} s^{-1}$, what is the ball’s amplitude after $30$ complete cycles?

(3) You are watching your friend practice diving, and you notice that, after your friend leaps off the board, the diving board continues to oscillate for a few seconds before stopping. You observe that the amplitude of oscillation of the board decays to 75% of its original value after 4 seconds. What is the ratio of the amplitude of the board after 6 seconds to the original amplitude?

Click HERE for solutions.

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

### Practice Problems

Conceptual problems

BoxSand's quantitative problems

Recommended example practice problems

- The Openstax section on Damping has only a few questions, Website Link
- Large set of practice exercises, the last section has problems on only damping, PDF Link

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.