What is anharmonicity and how does it relate to oscillation?

[Grevade]

For my physics lab report, we are supposed to conduct an experiment to show the non-harmonic oscillation of a simple pendulum.

I know what is simple harmonic oscillation, damped oscillation, driven damped oscillation. But what is a non-harmonic oscillation?
A google search reveals that there are also other forms of oscillation such as anharmonic oscillation and non-linear oscillation. Care to explain what are they and the differences.

Any idea what experiment I should conduct.

 

[jeppetrost]

Well, a simple pendulum only undergoes (approximate) SHO under certain condition, do you know which?

And by non-harmonic, I think you're simply supposed to show a pendulum -not- under SHO. Ie, you have to break the conditions you just thought of.

 

[vanhees71]

The pendulum obeys the exact equation of motion
$$m R \ddot{\varphi}=-m g \sin \varphi,$$
where $\varphi$ is the angular deviation of the pendulum from the direction of gravity ("down"). The above equation you can write as
$$\ddot{\varphi}=-\omega_0^2 \sin \varphi \quad \text{with} \quad \omega_0^2=\frac{g}{R}.$$
For small angles (i.e., small amplitudes of the motion) you can approximate $\sin \varphi=\varphi$. In this limit the solution of the equation of motion is harmonic (prove this!) and in this limit the frequency is $f_0=\frac{\omega_0}{2 \pi}$ is independent of the amplitude.

The exact equation cannot be solved in terms of elementary functions. The solutions are known as elliptic functions. They are also periodic as the sine and cosine function, but they are of course not simple sine and cosine functions. That's known as "anharmonic motion". The frequency for the exact equation depends on the amplitude of the pendulum's motion, and I guess you are supposed to measure the frequency as a function of the amplitude.

I'm not sure, whether you are supposed to evaluate the frequency numerically. This you can do by using energy conservation and solving for the period of the motion $T=1/f$, which gives you an elliptic integral, that cannot be solved in terms of elementary functions, but it's easy to evaluate numerically. If you need more help, don't hesitate to ask.

 

[rude man]

For my physics lab report, we are supposed to conduct an experiment to show the non-harmonic oscillation of a simple pendulum.

I know what is simple harmonic oscillation, damped oscillation, driven damped oscillation. But what is a non-harmonic oscillation?
A google search reveals that there are also other forms of oscillation such as anharmonic oscillation and non-linear oscillation. Care to explain what are they and the differences.

Any idea what experiment I should conduct.

I would measure period vs. amplitude. Demonstrate that the period is not constant with amplitude.

You can make a qualitative statement as to which direction the period should go (higher or lower) as the amplitude is increased. Think of what determines shm period of a pendulum and then ask yourself in which direction the period should go as the swing angle is increased to large angles like 45 deg.

 

[Grevade]

Well, a simple pendulum only undergoes (approximate) SHO under certain condition, do you know which?

And by non-harmonic, I think you're simply supposed to show a pendulum -not- under SHO. Ie, you have to break the conditions you just thought of.

Since SHO is only true for very small angles, so I guess we have to displace the pendulum at a large angle.

 

[Grevade]

I would measure period vs. amplitude. Demonstrate that the period is not constant with amplitude.

You can make a qualitative statement as to which direction the period should go (higher or lower) as the amplitude is increased. Think of what determines shm period of a pendulum and then ask yourself in which direction the period should go as the swing angle is increased to large angles like 45 deg.

In SHM, the period is independent of the amplitude, right? Wikipedia says it is isochronous.
For a non-harmonic motion, will the period increase as the amplitude increases? Take more time to cover a longer distance...

 

[vela]

Look at the equation of motion vanhees gave you. How do $\varphi$ and $\sin \varphi$ compare? Which corresponds to the greater magnitude restoring torque (at the same value of $\varphi$)? If you have a greater restoring torque, would you expect the system to oscillate faster or slower?

 

[Grevade]

The pendulum obeys the exact equation of motion
$$m R \ddot{\varphi}=-m g \sin \varphi,$$
where $\varphi$ is the angular deviation of the pendulum from the direction of gravity ("down"). The above equation you can write as
$$\ddot{\varphi}=-\omega_0^2 \sin \varphi \quad \text{with} \quad \omega_0^2=\frac{g}{R}.$$
For small angles (i.e., small amplitudes of the motion) you can approximate $\sin \varphi=\varphi$. In this limit the solution of the equation of motion is harmonic (prove this!) and in this limit the frequency is $f_0=\frac{\omega_0}{2 \pi}$ is independent of the amplitude.

The exact equation cannot be solved in terms of elementary functions. The solutions are known as elliptic functions. They are also periodic as the sine and cosine function, but they are of course not simple sine and cosine functions. That's known as "anharmonic motion". The frequency for the exact equation depends on the amplitude of the pendulum's motion, and I guess you are supposed to measure the frequency as a function of the amplitude.

I'm not sure, whether you are supposed to evaluate the frequency numerically. This you can do by using energy conservation and solving for the period of the motion $T=1/f$, which gives you an elliptic integral, that cannot be solved in terms of elementary functions, but it's easy to evaluate numerically. If you need more help, don't hesitate to ask.

Using the small angle rule,
$$\ddot{\varphi}=-\omega_0^2 \varphi$$
So we can say that the acceleration,$\ddot{\varphi}$ is directly proportional to the displacement, $\varphi$ and are oppositely directed. Is this sufficient to conclude that this is harmonic motion?

Is "anharmonic motion" and "non-harmonic motion" referring to the same thing?
So you are saying that, in anharmonic motion the frequency is dependent on the amplitude. So if I can show that the frequency changes as the amplitude changes, thus I have shown that it has undergone anharmonic motion.

So during the lab class, I displace the pendulum at a large angle and count the frequency of the pendulum, in let's say 10 secs. And then I repeat this for a range of amplitudes. I should get a graph that does not change sinusoidally. Any idea what I should expect the graph to look like? Am I right? Will it be the same if I measured the period for a number of oscillations instead of the frequency?

My lab report requires me to explain the physics theory in the choice of data collection and analysis. What is the physics theory? Do I explain that the pendulum only obeys SHM at small angles, so when displaced at large angles it will undergo non-harmonic motion.

How do I conclude that I have displayed non-harmonic motion?
Thanks for all the help.

 

[Grevade]

Look at the equation of motion vanhees gave you. How do $\varphi$ and $\sin \varphi$ compare? Which corresponds to the greater magnitude restoring torque (at the same value of $\varphi$)? If you have a greater restoring torque, would you expect the system to oscillate faster or slower?

I am guessing that for the same value of $\varphi$, $\varphi$ will be larger than $\sin \varphi$. So the SHM will have a larger restoring torque compared to the non-harmonic motion. Correct?
So if the SHM has a greater restoring torque, it will oscillate faster.
Does this means that at larger angles of $\varphi$, there will be a larger restoring force, so it will oscillate faster? Will the period be the same for different angles?

 

[vela]

I am guessing that for the same value of $\varphi$, $\varphi$ will be larger than $\sin \varphi$.

There's no need to guess. Just look at the graph of the two functions.

So the SHM will have a larger restoring torque compared to the non-harmonic motion. Correct?

Yes.

So if the SHM has a greater restoring torque, it will oscillate faster.

Yes.

Does this means that at larger angles of $\varphi$, there will be a larger restoring force, so it will oscillate faster?

As $\varphi$ increases, the approximation $\sin \varphi \cong \varphi$ gets worse. What does this imply about the behavior of the oscillator/pendulum?

Will the period be the same for different angles?

You should be able to answer this with a little thought.

 

[Grevade]

thanks for the help.
can someone explain the difference between non-harmonic, anharmonic and non-linear oscillation?

 

[vela]

Read http://en.wikipedia.org/wiki/Anharmonicity