Harmonic Oscillator¶
Textbook & Collegerama
- Chapter 4: sections 4.1 - 4.3
- TN1612TU_12 & _13
Learning goals
- List and describe types of oscillators
- Derive equations of motion for different types of harmonic oscillators
- Solve problems with mass-spring systems
The harmonic oscillator is probably the most important thing to remember from this class. That is because for small deviations from a stable equilibrium point all systems, irrespectively of their exact nature and what type of pertubation, behave as a harmonic oscillator (see Potential Energy).
In many situations, a particle or object is in some form of oscillatory motion. The physics of these motions usually leads to identifying a restoring force, i.e., a force that tries to drive the particle back to some equilibrium state and due to some form of inertia the motion that is currently present has the tendency to continue.
Why this somewhat abstract text? The reason is that outside classical mechanics, oscillating systems are also found. For example, the disturbance of an electromagnetic field to the electrons in a molecule, give rise to (driven) oscillations, the harmonic response of the electrons is called Raman scattering. It is convenient if you then can make an analogy with a classical mechanics system. It is usually easier to visualize, and its concepts are closer to daily life.
First, we will deal with the archetype of oscillations: the harmonic oscillator. The harmonic oscillator is a system that, when brought out of equilibrium, experiences a force that drives it back to equilibrium, with force proportional to the distance out of equilibrium.
Undamped harmonic oscillator¶
A concrete oscillator: Mass-Spring system. Let's look at a particle of mass
The figure shows the archetype: a mass on a spring is brought out of equilibrium by displacing it by
We set up the equation of motion for the particle. One force acts on it with the
with initial (or boundary) conditions: at
The solution of the harmonic oscillator equation of motion is a combination of sines and cosines (complex exponentials in general):
with
Instead of
Note: the dimensions of the constants
The equation of motion is a second order differential equation with constant coefficients. It is linear because all derivatives only appear linear, e.g. not
Note: a heavy particle (large
Notice that the oscillation amplitude does not change over time (the oscillation is undamped, see the figure)! In our daily lives, we do not see this much, if at all. Most oscillating systems sooner or later come to rest unless we provide some driving force (like the pendulum of a clock: if there is no energy source, the pendulum will come to rest). The reason is: that we have not included any damping effect ("friction").
Solution via complex exponentials¶
Later you will see that a combination of complex exponentials
Substituting into the equation of motion
But as both
Now we have the homogenous solution to the differential equation. To find the particular solution (i.e. a solution for specific boundary conditions) we need to include constants
This gives us for the two constants
We see that the constants are complex conjugate of each other
Of course this is the same solution as above, however this is the general solution strategy for linear differential equations with constant coefficients. NB: The solution is pure real, although we started with a general complex ansatz. This makes sense as the oscillation must have a real dimension. NB: remember
Harmonic Oscillator via the potential¶
The force driving of the oscillation is given by
Where we have taken
Thus the energy equation for the harmonic oscillator is:
Taking the time derivative of the energy equation gives the equation of motion, as we can divide out
Note that for all potentials that have a stable equilibrium point (
Damped Harmonic Oscillator¶
In many cases, an additional friction-like force acts on the mass. Consequently, some of the spring's kinetic and/or potential energy is converted into heat, and the oscillator's motion decays over time: the oscillation is damped.
A simple case occurs when the damping force is proportional to the velocity of the mass. The equation of motion includes a friction force, i.e.,
With initial conditions: at
The solution of the damped harmonic oscillator equation of motion is more complicated than that of the harmonic oscillator. We use the ansatz
with solutions (roots)
Depending on the respective values of
-
Case:
are complexThe solution is an oscillation with decaying amplitude over time. The oscillation frequency is always smaller than the natural frequency
. The characteristic time of the damping/decay of the oscillation is given by the constant in the exponent as . This expression has the unit [1/s]. -
Case:
both roots collapse and there is only one solution. The oscillation part of the solution disappears and only the decay remains.This case is called critically damped, because it is the fastest way to go back to equilibrium position. (Note: this has practical value is case you want construct e.g. a door closer. It should close as fast as possible, but without that the door hits the frame before).
-
Case:
both roots are realNot really oscillating. The first term dictates the long-term behaviour (the second one decays faster). This case is called over damped.
Driven Damped Harmonic Oscillator¶
Oscillators sometimes experience a driving force that can be periodic in itself. We will take here the case of a sinusoidal force with frequency
This gives for the equation of motion:
with initial conditions: at
The solution of the driven damped harmonic oscillator equation of motion for the case
With
The two other parameters
For
Evolution of the damping¶
Here we will have a quick look how the damping is evolving, that is we look at the root of the characteristic equation
and see how it evolves as a function of the damping
This gives quickly a qualitative view on the different regimes of the damping. The root
- No damping:
. The root is pure imaginary with two conjugate solutions on the imaginary axis. This gives pure oscillations. - Some damping
. The root is complex, with real and imaginary part, the oscillation will damp out over time (shown in blue, underdamped regime). . The roots collapse into one pure real root (critically damped), no oscillation.- Lots of damping
. The root splits into two real roots, no oscillations (shown in yellow, overdamped regime).
The root walks over the shown graph from
From this plot you can directly see that the system is stable for
Coupled Oscillators¶
In this course we mostly only consider one oscillator, but of course there could be many that are coupled in one way or another. Already Christiaan Huygens considered them.
There are 2 pendula suspended from a common connection, which rests on two chairs. If you set the pendula in motion, they will be initially out of phase, i.e. the relative position of the pendula is different. But over time their motion synchronises! What has happend? Apparently the two pendula are connected, coupled, via the suspension and act on each other, they are not independent, but influence the motion of the other pendulum.
In this movie, you can see a modern day version of this phenomena. Here the pendula are coupled via the ground. This influence is called weak coupling. In this course we cannot treat this coupling mathematically, but in the second year course on Classical Mechanics you will learn to study systems like these.
Examples¶
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Example of resonance: sound waves are exciting a glass. By changing the frequency of the sound waves to the resonance frequency, the glass starts oscillating with increasing amplitude until it finally breaks.
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Driven harmonic oscillator with damping.
Click on the image to start the physlet.
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1940: the Tacoma Narrows Bridge in the state Washington on the West coast of the USA is brought into resonance by the wind. The end result: click the movie to see it yourself.
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Breaking a HDD hard disk with a song of Janet Jackson
Read here about this truly amazing piece of applied physics on a blog of Microsoft developer Raimond Chen.
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Raman scattering.
If electromagnetic radiation interacts with a molecule to zero order, then the molecule does not vibrate or rotate. The radiation with field
then induces a dipole moment in the electron cloud of the molecule what will vibrate with the same frequency as the external field . This phenomena is called Rayleigh scattering.If the electrons in the molecule already oscillate with another frequency
, and the polarizability of the molecule depends on the distance from the nuclei of the molecule, to first order the polarizability can be modeled as . The emitted field is thereforeWe see that next to the Rayleigh term
two additional lines with frequency appear. These lines are called Raman lines. They are characteristic for the vibrational bands of the molecule present and are used to probe the presence of certain molecules in practice. NB: The line with frequency has higher energy than the incoming light. This line has a much smaller intensity than the line. To understand this you need to consider quantum mechanics and that the occupation of states is Boltzmann distributed (but this is beyond this example and you will learn more about this in the second and third year).We hope that you see the similarity to the harmonic oscillator where the polarizability
is modelled as a (restoring) force. -
Second-harmonic generation
Of course the harmonic potential is only a first order approximation around an equilibrium. An example, for a non-linear force or anharmonic potential effect, is the generation of second-harmonic generation. If you shine high intensity light onto the electrons of a molecule, they are pushed out of equilibrium further and if the governing potential is anharmonic, the electric field response will not only include the incoming frequency
but also higher harmonics , but with much lower intensity. That the emitted frequencies are occurring in integer multiple of the incident frequency can be understood either from quantization of light into photons (and the conservation of energy) or from Fourier analysis of the periodic motion of the electron. -
Erasmus Bridge & singing cables.
The bridge in Rotterdam, but also others, suffer from long cables that the wind can put into resonance. Their motion then generates acoustic waves in the audible spectrum. Listen here to the sound of the cables starting from 1:00 on the website for singing bridges!
Exercises¶
Here are some exercises that deals with oscillations. Make sure you practice IDEA.
-
A massless spring (spring constant
) is suspended from the ceiling. The spring has an unstretched length . At the other end is a point particle (mass ).-
Make a sketch of the situation and define your coordinate system.
-
Find the equilibrium position of the mass
. -
Set up the equation of motion for
. -
Solve it for the initial condition that at
the mass is at the equilibrium position and has a velocity .
-
-
Same question, but now two springs are used. Spring 1 has spring constant
; spring 2 has . Both have the same unstretched length .-
The two springs are used in parallel, i.e., both are connected to the ceiling, and
is at the joint other end of the springs. -
Both springs are in series, i.e., spring 1 is suspended from the ceiling, and the other one is attached to the free. The particle is fixed to the free end of the second spring.
-
Do it yourself¶
Find a rubber band and use nothing but a mass (that you are not allowed to weigh) that you can tie one way or the other to the spring, a ruler, and the stopwatch/clock on your mobile.
Set up an experiment to find the mass
Don't forget to make a physics analysis first, a plan of how to find both
Jupyter labs¶
- Mass-spring system Exercise4.ipynb