Harmonic vibration – all about the topic

The harmonic vibration

This article is about harmonic oscillation. We explain to you what harmonic oscillation is, derive its mathematical description and also show you its meaning using an application example.

This article belongs to the subject Physics and represents a subtopic of the topic Oscillations.

Harmonic Vibration – What is it?

As a reminder: An oscillation is generally a periodic change in one or more physical quantities in a physical system. Since various disciplines deal with the topic of vibration, we will deliberately limit ourselves to treating them within mechanics. Harmonic vibrations are also mechanical vibrations in which a body regularly moves around an equilibrium position (rest position). If the path-time function of a mechanical oscillation also has the form of a sine function, then we call it harmonic, otherwise aharmonic.

In the following, we shall now be concerned precisely with those harmonic oscillations or movements. But how do we derive the equation of motion for such?

Derivation of the equation of motion for harmonic oscillations

In order to find a function for the deflection (elongation) as a function of time, we make the following consideration:

The projection of a uniform circular movement corresponds to the movement of a harmonic oscillator. By that we can imagine the movement of a body on a circular path, in which equal distances are covered in equal time segments. For us it is particularly important to know that the amount of the orbital speed remains the same, but not the direction.

The radius r corresponds to the amplitude ymax and the period of rotation corresponds to the period of oscillation t:

Fig. 1: The projection of a uniform circular motion

from: https://physics-lessons-online.de/

The following applies to the elongation y:

The angle (phi), also known as the phase angle or just phase, can be expressed using the orbital period. Because it applies:

For an entire revolution or a complete oscillation process (i.e. for the period T) also applies:

The quotient 2T is called the angular frequency or angular velocity (omega):

With this one can also write for the phase angle:

The following therefore applies to the time profile of the deflection y:

For uniform circular motion, the angular frequency is constant.

So it applies

So we have found a function for a harmonic oscillation that corresponds to the deflection y as a function of the time t. It is:

We can call this function the equation of motion for harmonic oscillations.

Equation for harmonic oscillations

The equation for harmonic oscillations can also be expressed using the oscillation period T or the frequency f.

To do this, you replace the angular frequency with again

So you can express the equation for harmonics in different ways:

Addition: All oscillating systems are called oscillators. Oscillators whose path-time function corresponds to a sine function are called harmonic oscillators.

Relevance of the harmonic oscillation equation

Now the question arises what we can do with the oscillation equation. The answer to this is that given a known period or frequency and a known amplitude, we can calculate the excursion of a harmonic oscillator at any time t.

Depending on which of the quantities , T or f is known, we select one of the three variants of the oscillation equation mentioned above.

Application example for the harmonic oscillation equation

A harmonic oscillator oscillates with an oscillation period of 1.2 seconds. The maximum deflection is 12 cm.

At time t = 0 s the oscillator is in the rest position on the way up in the positive y-direction.

Question: Where is the oscillator at the following points in time?

  1. t = 0.6s
  2. t = 1s
  3. t = 1.5s

Solution:

The following values ​​are given:

T = 1.2s

y max = 12 cm

We insert the given values ​​into the oscillation equation for harmonic oscillations and thus calculate the respective deflection.

y