Second Order ODE's: Undamped forced oscillations: Beats

In my previous post, we discussed the principles of mechanical resonance. In this post, we'll look into another vibratory characteristic called "beats", which can also be described by our undamped, mass-spring model (Figure 1).

Figure 1. Undamped, forced oscillation mass-spring system.

What are beats?

Basically, beats occur due to the interference of two sound waves of slightly different frequencies. If you're a musician, you can hear beats when tuning instruments. If a note is out of tune, you will hear "wavering" or "fluttering" of the note. These are the beats.

Have a listen to the guitar being tuned via the link below. Two strings are plucked simultaneously (D string 5th fret; open G), but are slightly out of tune. Can you hear the beats?

Click here to listen to Guitar Tuning (source: WikiMedia Commons)

Beats can also occur when we excite the mechanical system (Figure 1) with vibrations that are close to its natural frequency. And this is what we're going to discuss in this post.

Let's look into the mathematics of beats in regards to our mass-spring system...

Beats

Equation (4) of the previous post (post #14) describes the general solution of the undamped mass-spring system. We'll label it as equation (1) here...

Now suppose that the conditions of the system are such that δ = 0 and x(0) = 0, then we find...

Thus the particular solution reduces to...

By the sum to product formulas (trigonometric identities), the term in the brackets can be expressed as...

Substitute this into equation (2) and the particular solution becomes...

Equation (3) is the mathematical representation of beats.

So what occurs physically with our system?

When the input frequency (excitation frequency) ω is close to, but not quite equal to the natural frequency ω₀, the mass m will bob up and down in the manner depicted graphically in Figure 2 below.

Figure 2. Response of the mass-spring system

Here's what I find fascinating about beats: the waveform is enveloped between the two large blue dashed sine waves. This sine waves have a frequency that is half of the difference between the natural frequency and excitation frequency...

This is called the envelope frequency.

The frequency of the response (or sound) produced is the average of the natural and excitation frequencies...

We see that within one cycle (one period) of the enveloping sine wave, there are 2 peaks in the modulating amplitude of the response. This is what we hear as beats. Thus the beat frequency is twice the envelope frequency, or...

...which is simply the difference between the natural and excitation frequencies.

In the next post, we will see how the inclusion damping affects the response of the oscillating mass. Stay tuned.

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Credits:

All equations in this tutorial were created with QuickLatex

Graphs are produced in desmos.com/calculator

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2. Differential Equations: Order and Linearity
3. First-Order Differential Equations with Separable Variables - Example 1
4. Separable Differential Equations - Example 2
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Second Order Differential Equations

1. Introduction to Second Order Differential Equations
2. Finding a Basis for solutions of Second Order ODE's
3. Roots of Homogeneous Second Order ODE's and the Nature their Solutions
4. Modelling with Second Order ODE's: Undamped Free Oscillations
5. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 1
6. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 2
7. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3
8. Non-homogeneous Differential Equations
9. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 1
10. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 2
11. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 3
12. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 4
13. Modelling Forced Oscillations with Second Order ODE's
14. Second Order ODE's: Undamped forced oscillations: Resonance
15. Second Order ODE's: Undamped forced oscillations: Beats

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