There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis

Monday, March 28, 2011

Shock excitation

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Simple Resonators - Shock excitation

2.1 The decay equation

One way to excite a resonance is with a shock, such as the striking of a bell, hitting a tuning fork, plucking a violin string or the exciting of any resonator with a suitable impulse. The resonator then oscillates in a decaying fashion as shown in the animation below. Click on it to start it.

Figure 1. The bell ringer. Click on the figure to start or restart it. Mousing off will stop it (the sound requires a restart click to restart it). Real bells oscillate much, much faster than is indicated here and usually resonate at more than one frequency.

The equation for this decaying oscillation is:

the basic decay equation of an elemental resonator    .   (1)

This is the decay equation. The decay equation is the topic of this posting.

The decay equation is written, in this case, in terms of the acoustic pressure p(t), the small oscillating pressure that our ears tell us is sound. The quantity τ is the time constant of the decay and A is the amplitude at the starting time, i.e. at t = 0 where t is time. Mathematically, e raised to the time dependent power −t/τ is responsible for the decaying amplitude while the cos(ωt) mathematically causes the oscillations. The mathematical constant e (equals 2.718... ) is the base of the natural logarithm.

You can further understand the two parts (factors) of Equation (1) as follows:

Table 1. Parts of Equation (1) About exponential functions
equation for the envelope      (1a)

The first factor, the decaying amplitude, is an exponential function.

It also serves as an envelope for the second term. In electronics, an envelope is a curve that bounds another curve.

The envelope can be seen in the above animation by clicking on "show envelope".

It is also shown in Figure 2b below.

Figure 2b below shows an exponential function such as Equation (1a). Such a function is characterized by a time constant. The time constant, τ, is the time the function requires to fall to 1/e (i.e. 0.37 or 37%) of its its initial value, as is shown in Figure 2b with the blue dots. One amazing thing about this function is that you can use any point along the curve as the "initial" point, and one time period later the function will equal 37% of this "initial" value. That is to say, it is a constant of the function, just like the slope of a straight line is the same no matter where it is measured.

Another interesting feature of this function is that it becomes a straight line if the scale on the vertical axis is logarithmic, as is shown in Figure 2d. Investors often use this type of graph for plotting the value of various investments. An investment that is rising (or falling) at a constant percent rate of return will be exponential and therefore straight on this type of graph. It is also true that the slope of the straight line will be proportional to the rate of return, making of this type of graph good for quick identification of the rate of return of investments.

Some fields of study use half-life t½ to characterize exponential decay in place of the time constant. This is the time this function takes to fall to half its "initial" value. The half-life and time constant are related as t½ = τ ln(2) = τ×0.693 .

Exponential functions are the solution to a simple first order differential equation and define the behavior of many scientific phenomena. See these sites for discussions of exponential growth and exponential decay.

cos(ωdecayt)  (1b) The second factor is the constant oscillating sinusoid. A sinusoid is any sine or cosine function or sum of the two that have the same frequency, ω. A typical sinusoid is shown in Figure 2c.

graph of decaying oscillations graph of decaying exponential envelope constant sinusoid semi-log graph of decaying exponential
Fig. 2a. The whole waveform, a decaying sinusoid as given by Equation (1). We break this up into two factors in Figures 2b and 2c. Fig. 2b. The first factor: decaying envelope as given by Equation (1a). This is a simple exponential function with decay time constant, τ. Fig. 2c. The second factor: a continuous ongoing sinusoid as given by Equation (1b). Fig. 2d. An alternate way to graph an exponential function, with a logarithmic axis. The exponential function is now a straight line.

Exponential decay is the best known type of decay for simple resonators; however, some resonators decay differently. For example, if sliding friction dominates in a resonator (such as between two rubbing parts), then the decay (i.e. the envelope) may be linear, allowing the resonator to totally stop in a finite time, instead of just asymptotically approaching zero, as in Figures 1 and 2a. Other types of decay would be governed by different differential equations, primarily different in the term that involves the source of the decay. We will continue to focus on exponential decay in this posting.


photograph of LRC circuit under test
Figure 3. Photo of an LRC circuit with test leads (wires) connected. In most commercial electronics, these components are much smaller in size than those shown here, are attached to a circuit board, and are part of a larger electrical circuit.

See Figure 4 for an illustration with the test instruments also in place. Most students doing this experiment would attach these components to a prototyping board to better hold them in place. We did not do that here in order to better show the naked LRC circuit components. The red handled clip at the top left serves to hold two wires together.

LRC circuit resonators

We next explore a second resonator to complement the example of the bell above. This second resonator is the LRC* circuit resonator. The LRC circuit is a typical linear resonator which was presented in the last posting. We will use this circuit to derive the basic decay equation presented above (Equation (1) ). We picked the LRC ciruit because of its widespread use in all sorts of electronics, because it is simple, and it is often presented in text books as being representative of all simple resonators. It sees use in filters, matching circuits, and oscillators in many types of electronic devices. The resonant frequencies of LRC circuits can range from tens of Hertz (for very low audio signals) to GHz for computers, cell phones, and GPS devices. See my posting on Marconi and the spark gap transmitter as an example of an historic use of an LRC resonator.
*Standard text books vary on the naming of this circuit. LRC, RLC, and LCR are used.

An LRC circuit is shown in the photograph at the right in Figure 3 and consists of an inductor, a resistor, and a capacitor in series. The inductor is basically a small coil that provides a "momentum" to the current flow. The capacitor has two plates separated by an insulator (non-conductor) and stores electrical charge, similar to the Leyden jars that Benjamin Franklin used. The resistor is a circuit element that provides some electrical resistance to current flow and in the case of an LRC resonator, it provides damping to the oscillations.

The animation below shows a typical test setup used to excite and observe the decaying oscillations of this resonator. On the left of this figure, we see a pulse generator used to shock-excite the resonator (in the middle). On the right is an oscilloscope used to observe the oscillations in the resonator. Click on the figure to start the animation and mouse off to stop it. Note that in student electronics labs, a general purpose signal generator, set to make square waves, often replaces the pulse generator, with the resulting signal being slightly more complicated but still generally similar to the result shown here. The oscillations are usually too fast for the eye to follow, but the oscilloscope captures them for viewing.

Figure 4. Test setup for monitoring the resonance of an LRC circuit using shock excitation and decay. After the pulse, the pulse generator acts as a simple wire connecting points 1 and 2. This animation shows symbols in place of the actual inductor, resistor, and capacitor, shown in Figure 3. The pulsating shading inside the inductor and capacitor is meant to illustrate the invisible oscillating magnetic and electrical energies in these components.


diagram of LRC circuit resonator

Mathematical derivation of the decay equation

In this section we will show a derivation of the decay equation, Equation (1).

For this derivation, we will use the LRC series circuit as our example elemental resonator. As was discussed in the last posting, all the elemental resonators have equivalent defining differential equations and will behave in a similar fashion. This circuit is shown at the right and is made up of an inductor L, a resistor R, and a capacitor C in series as shown in Figures 3 and 4 above. The same oscillating current I(t) flows through all three elements, while the voltages across the three elements differ from each other. The oscillating charge q(t) is the charge on the upper plate of the capacitor C (the lower plate has an equal magnitude but oppositely signed charge).

Table 2. We use the following elemental equations for circuits:
VC = q/C   (2a) The voltage across the capacitor, where C is the capacitance and q is the charge on the upper capacitor plate.
VR = IR   (2b) The voltage across the resistor as given by Ohm's law. R is the resistance of this resistor.
VL = L dI/dt   (2c) The voltage across the inductor. L is the inductance of the inductor used.
zero total voltage   (2d) After the initial shock (that starts the oscillations), the sum of all the voltages is zero.
current from capacitor   (2e) The current flowing through all circuit elements (the L, the R, and the C) equals the time rate of change of charge on the upper capacitor plate.

Table 3. Combine the above in the following steps:
differential equation in terms of q   (3a) Substituting Equation (2e) into the Equation (2d), we get the defining differential equation for the behavior of this circuit. This is the differential equation we listed in Figure 1 of Section 1 for the LRC circuit, but here we have actually derived it!
decay equation in terms of q   (3b)
Equation (3a) is a very standard second order differential equation with known solutions of the form of Equation (1). In Equation (3b), p in Equation (1) has been replaced with q which is the oscillating variable in Equation (3a). To be completely correct, the cosine factor could have an arbitrary phase associated with it, i.e. cos(ωdecayt) ⇒ cos(ωdecay + φ) , where φ is the phase shift. At the same time, φ is not needed in the case at hand (it equals zero) so we leave it out for simplicity.
Equation (3b) is our decay equation! We are finished! We have derived the decay equation!


Derivation of constants for Equation (3b)

We will next derive equations for the constants τ and ωdecay in Equation (3b) above. This effort will prove to be more work than deriving Equation (3b).

Table 4. Derivation of constants in Equation (3b)
first derivative of decay equation   (4a) The first derivative of q(t). We will need this to substitute into Equation (3a).
second derivative of decay equation (4b) The second derivative of q(t). We also will need this to substitute into Equation (3a).
form of resulting equation   (4c)
Substituting Equations (3b), (4a) and (4b) into the Equation (3a), we get an equation with six terms. These can be sorted into sine and cosine terms to give an equation of the form shown here, where the constants B and C are shown below in Equations (4d) and (4e).
cosine term   (4d)     sine term   (4e)

Table 5. Now we're ready for some results (in our pursuit of the constants)!
time constant for the decay   (5a) Let's look at Equation (4c). As time progresses, the sine and cosine terms rise and fall out of phase with each other. In order for Equation (4c) to be valid for all times (i.e. the left side equaling zero at all times), both the sine and cosine coefficients need to be zero. Setting C=0 in Equation (4e) and solving for τ gives an equation for the time constant, τ (with units of seconds).
angular resonant frequency equation   (5b) Setting B=0 in Equation (4d), solving for ωdecay, and using Equation (5a), gives an equation for the shock-excited angular resonant frequency (or decay angular resonant frequency). ω0 is defined in the next equation.
angular resonant frequency for continuous excitation   (5c) ω0 is the angular resonant frequency for continuous excitation (discussed in a future posting) as opposed to shock excitation being discussed here. The effect of the decaying amplitude is to slightly reduce the effective frequency of the oscillations. (Perhaps this is easiest to understand in a mechanical resonator like the pendulum: drag will slow down the bob on its swings.) The units for both ωdecay and ω0 are radians per second.
approximate angular resonant frequency   (5d) In the case of small damping, i.e. when τ>>LC, then ωdecay approximately equals ω0 and the distinction between ωdecay and ω0 goes away.
resonant frequency equation   (5e) The decay resonant frequency in Hz (or cycles per second) equals ωdecay divided by two pi. You can treat this as a simple unit conversion if you wish: conversion of angular frequency to frequency
voltage across the capacitor as a function of time   (5f) In doing the experiment shown in Figure 4, we usually observe the voltage across the capacitor on the oscilloscope screen and not the charge. We use Equation (2a) above to convert our q(t) from Equation (3b) into VC(t), where τ and ωdecay are given by Equations (5a) and (5b).

If we should instead wish to know the voltage across the resistor or inductor, then we could similarly have used Equations (2b), (2c), and (2e) to convert our q(t) into these quantities as a function of time. Be warned however, the algebra for VR(t) and VL(t) is more involved than that shown here for VC(t). It might be easier to use complex methods to determine these voltages as discussed in the next posting. We would expect to get a phase shift φ as mentioned to the right of Equation (3b).

All the quantities, q(t), VC(t), VR(t), and VL(t), behave in very similar fashions: all are sinusoidal oscillations with exponential decaying amplitudes and look like the function in Figure 2a. All have the same values for τ and ωdecay.


Table 6. General results for all simple resonators
general second order differential equation   (6a) The various simple resonators shown in the previous posting were all governed by differential equations similar to that of the LRC circuit shown in Equation (6a). In the rest of this table we show the results when the derivation is done in terms of these general coefficients instead of those specific for the LRC circuit. This gives us results that are applicable to this whole class of resonators. I should point out that in this table C is not the capacitance, but instead it is the constant in the last term in Equation (6a).
Predictably, all these simple resonators respond to shock excitation in very similar ways:
general exponential decay equation (6b) All respond with a decaying sinusoidal oscillation of the form of Equation (6b), where q(t) could be replaced by any of the equivalent oscillating displacement parameters of a simple resonator, shown in Figure 2 of the last section. A1 is the amplitude of the parameter at t=0.

Because the kinetic and potential parameters of Figure 2 of the previous section are proportional to the derivatives of the displacement parameters they will also be of the same form as Equation (6b) with perhaps a phase shift φ in the cosine factor as noted with regard to Equation (3b) above. The proportionality is easier to understand in the complex notation of the next posting, Equations (6e) and (6f) )

τ = 2A/B   (6c) All have decay time constants τ given by (6c)
general decay angular resonant frequency   (6d) All have decay angular resonant frequencies given by (6d),
general constant excitation angular resonant frequency   (6e) Equation (6e) defines the ω0 in Equation (6d). Again, in this equation, C is not the capacitance, but just the constant of the last term of the general differential equation (6a).

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up: Contents
next: 2.2 Complex math derivation
P. Ceperley, Jan. 2010.