Second Order ODE's: Undamped forced oscillations: Resonance

Figure 1. Sound shatters a wine glass
(source: YouTube, creative commons licence)

In engineering and acoustics, second order differential equations can give us a fascinating insight into mechanical vibrations. In my previous post (post #13), I stated that it is important that the response be expressed in terms of the natural frequency of the equivalent undamped system. Now we find out why...

Figure 2. Undamped, forced oscillation mass-spring system.

There are 2 vibrational phenomena that we can study with a simple, undamped forced oscillating mass-spring system:

1. Resonance: engineers must avoid this when designing structures
2. Beats: the tremolo sound you can hear when 2 musical instruments are slightly out of tune.

Let's take a look at resonance first...

Resonance

In an undamped mass-spring system (Figure 2), with the damping coefficient c = 0, the equation of motion reduces to...

where...

...is the natural frequency of the undamped system.

Now, by equation (3) of post #4, the solution to the homogeneous ODE of (1) is...

From the final equation of post #13, the particular solution of the non-homogeneous ODE reduces to...

Therefore the general solution to equation (1) is...

Now, if we let the amplitude of x_p be A₀. Thus...

The greek letter, ρ, or rho, is called the "resonance factor". Here we see that ρ is very much dependent on the input frequency ω.

As the input frequency approaches natural frequency...

...we see that ρ, and the amplitude becomes infinitely large...

This phenomenon of excitation of large amplitudes by matching input and natural frequencies is called resonance.

Let's now illustrate resonance graphically, by matching the input and natural frequencies into the equation of motion. So equation (1) becomes...

For equation (5), since x_p is also a solution of x_h, equation (4) is no longer valid. By the modification rule (post #12) we need to choose x_p to be of the form...

Therefore...

...and...

Sub these into (5) and we get...

Now equating coefficients...

Thus...

Figure 3 below illustrates equation (6), the response of the mass-spring system under resonance.

Figure 3. increasing amplitude of oscillations of the mass, m

We see that as time progresses on, the amplitude of the oscillating movements of the mass m increases linearly. If we don't control this increase in amplitude, the system will eventually experience a catastrophic failure.

A great example of such a catastrophic failure due to resonance is the opera soprano shattering a wine glass with a sustained, high pitched tone (Figure 1). The wine glass shatters because the note sung happens to be at a frequency that is equal to the resonant frequency (i.e. the natural frequency) of the glass.

Another great example is the infamous collapse of the Tacoma Narrows Bridge in 1940 (Figure 4), where eddy currents caused by winds passing over the bridge matched the resonant frequency of the structure, resulting in the bridge twisting and wobbling up and down violently until structure gave way.

This is a key reason why engineers must design structures with its resonant frequency in mind. They must ensure natural and artificial events cannot excite a structure at its resonant frequency.

Figure 4.Tacoma Narrows Bridge collapse (source: WikiMedia Commons)

In the next post, we'll look into the mathematics of the phenomenon of "beats".

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Credits:

All equations in this tutorial were created with QuickLatex

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First Order Differential Equations

1. Introduction to Differential Equations - Part 1
2. Differential Equations: Order and Linearity
3. First-Order Differential Equations with Separable Variables - Example 1
4. Separable Differential Equations - Example 2
5. Modelling Exponential Growth of Bacteria with dy/dx = ky
6. Modelling the Decay of Nuclear Medicine with dy/dx = -ky
7. Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky
8. Mixing Salt & Water with Separable Differential Equations
9. How Newton's Law of Cooling cools your Champagne
10. The Logistic Model for Population Growth
11. Predicting World Population Growth with the Logistic Model - Part 1
12. Predicting World Population Growth with the Logistic Model - Part 2
13. What's faster? Going up or Coming Down?

First order Non-linear Differential Equations

1. There's a hole in my bucket! Let's turn it into a cool Math problem!
2. The Calculus of Hot Chocolate Pouring!
3. Foxes hunting Bunnies: Population Modelling with the Predator-Prey Equations

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

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