Sensors - May 2004 - An Introduction to Classical Sinusoidal Vibration

Sinusoidal vibration is the simplest and easiest to understand of all the forms of vibratory motion. So don’t let the math scare you.

Figure 1. A parked vehicle undergoes no change of state in terms of displacement, velocity, or acceleration. The curves would look like these.

What is sinusoidal vibration? Sinusoidal (or sine) vibration smoothly varies with time. Let’s first review a few terms that pertain to unidirectional motion and then extend them to sinusoidal vibration. Visualize your automobile, parked at curbside (see Figure 1). The engine may be running, but the car is not moving. The odometer (distance or displacement) is not changing with time. The speedometer (velocity) indicates zero. There is also zero acceleration. Now, if we smoothly increase our velocity by holding our acceleration constant, the distance or displacement will increase proportional to time. The velocity graph in Figure 2 plots the slope, the rate of change of displacement. To go from the displacement graph to the velocity

Figure 2. With an increase in velocity at constant acceleration, displacement increases proportionally to time squared.

graph, we differentiate (take the slope of) the displacement graph. The acceleration graph plots the slope, the rate of change of velocity. To go from the velocity graph to the acceleration graph, we differentiate, take the slope of, the velocity graph. Acceleration is thus the second derivative of displacement.

Visualize performing an automotive experiment and generating plots like those in Figure 2. You might measure acceleration, deriving a signal from an accelerometer. You could electronically integrate that signal once to get a signal proportional to velocity, and then again (now “double integration”) to get a signal proportional to displacement. Note that each time we differentiate, we plot the slope of the graph above. Each time we integrate, we plot the area under the curve below.

Figure 3. A passive single differentiator uses a signal from a velocity sensor to obtain an acceleration signal.

Here’s a situation in which we would differentiate electrically: Using a signal from a velocity sensor, we could differentiate to obtain an acceleration signal (see Figure 3). This is precisely what was done back in the 1950s, before piezoresistive and piezoelectric accelerometers were developed.

Visualize a constant-voltage rising-frequency sine wave applied to the “velocity” terminals of what is, in effect, a high-pass filter. The output at the “acceleration” terminals will rise at 6 dB/octave, precisely what we need. Or we could electrically integrate the signal from that velocity sensor to obtain a displacement signal (see Figure 4).

Figure 4. A signal from the velocity sensor can be integrated to obtain a displacement signal.

Now visualize a constant-voltage rising-frequency sine wave applied to the “velocity” terminals of what is, in effect, a low-pass filter. The output at the “displacement” terminals will diminish at –6 dB/ octave, precisely what we need.

If we begin with a signal from an accelerometer, we will integrate once to obtain a velocity signal. Most of today’s velocity sensors are no longer massive coil-and-magnet devices but rather quite small, light accelerometers with an internal microchip integrating amplifier. Active integration (see Figure 5) permits integration over a wider range of frequencies than does passive integration.

Figure 5. Active integration techniques can be used over a wider range of frequencies than is possible with passive integration.

The functions of these analog integration circuits are nowadays performed digitally.

If you, the reader, were operating a shaker, performing a vibration test, your shaker-control sensor would be called the control accelerometer. You would double-integrate its signal (see Figure 6) in order to regulate and display vibratory displacement.

Figure 6. In vibration tests, acceleration signals are double-integrated to regulate and display vibratory displacement.

Sinusoidal Vibration
Now we are ready to view one-frequency-at-a-time sinusoidal vibration (see Figure 7).

Figure 7. When displacement, velocity, and acceleration are plotted against time they are called time histories.

Note that we plot displacement, velocity, and acceleration as before. Because all three are plotted against time, they are often called “time histories.”

Let X be the peak or maximum value of displacement, V represent the peak or maximum value of velocity, and A represent the peak or maximum value of acceleration. Note that when displacement reaches its extreme or peak values, velocity is zero. Note too that at any moment when displacement is zero, it is changing the most rapidly. Lay a straightedge onto the displacement graph of Figure 7. The straightedge is maximally steep; velocity is maximum + or – and “leads” displacement by 90°; and acceleration “leads” velocity by 90° and is 180° out of phase with displacement.

Figure 8. Displacement vs. time can also be plotted in degrees of rotation. The reciprocal of time, T, is the frequency, f.

The time required for one cycle is given as T and called the period. The reciprocal of T is the frequency, f.

But scientists typically denote their angles in radians (see Figure 9) rather than the degrees in Figure 8.

Figure 9. Scientists typically denote angles in radians rather than degrees. A radian is the angle (~57°) subtended by the radius if the radius were to be wrapped around the circumference of the circle.

A radian is the angle (~57.3°) subtended by the radius, if the radius were wrapped along the circumference of the circle.

radians = 180°, or one-half rotation; 2

radians = 360°, or one full rotation, etc.

The sine waves of these figures can represent varying displacement (first swinging +, then –) about some neutral position. Or varying air pressure swinging around one atmosphere. Or the electrical signal from a microphone exposed to sound passing along a tube. Pressure maxima cause a + signal, and minima a – signal. The sine wave suggests a pure tone, i.e., only one frequency present.

Now look at Figure 10, in which the vertical lines represent sound waves and can be visualized as the individual coils of a Slinky.

Figure 10. To envision sinusoidally varying air pressure, think of the vertical lines as sound waves. A compression wave, represented by the closely spaced lines, is followed by a rarefaction wave as air molecules pass energy to neighboring molecules.

The coils don’t move very far as they transmit energy from one to the next. A compression wave (shown as closely grouped turns) travels left to right, followed by a rarefaction wave. This represents air molecules in the duct passing energy to their neighbors. In the audible frequency range this is called “sound.” Longitudinal, or axial, waves also occur in structures, and are one form of vibration.

How fast does sound travel in one second? We call this the velocity of propagation (feet per second or meters per second) and assign it the symbol C. How far apart are the maxima? The minima? This is the wavelength and its symbol is the Greek letter lambda,

, measured in inches, feet, meters, or fractions of meters. These two terms relate to frequency thus:

So velocity = distance from maximum to maximum × number of such cycles in a second. It’s ~344 m/s, or 1127 fps, or 768 mph in dry air at sea level at 20°C. So let’s insert 344 m/s for C and 400 Hz for f. We calculate that

of a 400 Hz tone will be 0.86 m. The speed of sound in dry air at 20°C is 344 m/s. In water, it travels much faster and in solids, especially dense metals, yet faster. Figure 11 summarizes these concepts and shows you that with rising frequency, the wavelength gets smaller.

Figure 11. The velocity of propagation (C), or the speed of sound in a particular medium, is expressed as C =

× f, where

is the wavelength and f is the frequency. As illustrated here, the wavelength diminishes as the frequency rises.

Constant-Frequency Mathematics

Figure 12. The relationship among displacement, velocity, and acceleration can be understood in trigonometric terms by rotating line c to a point over A and thence around a circle several times.

Enough physics. Now a little math. In Figure 12, remember that the sine of angle BAC is equal to a/c, while the cosine is equal to b/c. If angle BAC grows, then sin BAC grows, while cos BAC shrinks. Visualize line c starting at 0°, rotating to a point directly over point A (90°), and continuing around a 360° circle not just once but many times. Radius c = AB is constant, but length a will change from zero to maximum (equal to c) then fall to zero and commence rising again (but in a negative or downward direction) to a negative maximum (again equal to c) and then fall back to zero. This sequence repeats with each rotation. Length a can represent an instantaneous voltage, which in turn can represent an instantaneous distance or displacement, x (Figure 7), or an instantaneous pressure (Figure 10).

Referring again to Figure 7, we could represent the upper trace of instantaneous displacement (or distance from zero) as x.
x varies with time according to:

Let us now differentiate, or take the slope or rate of change of x and call the result our instantaneous velocity v. See the second graph of Figure 7, with the instantaneous value:

Better yet, since most metrologists use peak-to-peak displacement D rather than zero-to-peak X:

We seldom use v. But we do need V, its maximum value. We select the instant when v = V, so that:

This is the “peak” or “vector” velocity, the greatest v value during any cycle.

International System of Units Example. Let f = 100 Hz and let D = 2.54 mm. Then:

To get an equation for a, the instantaneous acceleration (lowest trace of Figure 7), we differentiate or get the slope of our earlier equation:

We seldom use a. But we do need A, its maximum value. We select the instant when a = A, so that:

How do we evaluate that acceleration? We compare it with a widely recognized acceleration standard (the Earth’s gravitational acceleration at sea level), namely 32.2 fps² or 386 ips². Thus, A = 19,739/386 = 51.1 g. We can simplify the foregoing by using:

When we compare that with the widely recognized acceleration standard (the Earth’s gravitational acceleration at sea level), namely 9807 mm/s², we get:

In addition to the above relations among f, D, and A, you might need relations among f, V, and A. So you divide the A, f, D equation by the V, f, D equation:

All this math can be bypassed by using the cardboard calculators commonly given away by shaker and accelerometer manufacturers, but it’s a better idea to learn the equations and use a calculator to solve them. Or you can visit manufacturers’ Web sites and take advantage of their online and sometimes downloadable math.

Swept-Frequency Mathematics

Figure 13. If the frequency is rising uniformly with time, and displacement is held constant, the peak velocity increases linearly.

In Figures 7–10 we have been assuming that frequency remains constant, along with X, D, V, and A. Now let’s assume that frequency is rising uniformly with time, as in a “sine sweep” vibration test. D is being held constant, per many test specifications. Observe in Figure 13 that peak velocity V is increasing per:

Figure 14. Alternatively, if velocity is held constant, peak displacement diminishes with time and acceleration rises.

Alternatively, we might assume that V is to be held constant (see Figure 14). That would require peak displacement X to diminish with time. D would drop per:

Or we might assume that A is to be held constant, following, perhaps, some sinusoidal testing specification (see Figure 15). This would require that peak-to-peak displacement D would decrease proportional to rising frequency squared per:

Figure 15. Finally, if acceleration is held constant, peak-to-peak displacement decreases proportionally to rising frequency squared while velocity decreases linearly.

(22)

Suppose you were conducting a vibration test at 10 g peak at 1000 Hz. What would be the displacement D? Use Equation (22) to find out. An onlooker could not see or feel shaker motion and might even question whether the shaker was doing anything to the test article. You could correctly assure your visitor that if the test article had a natural frequency at 1000 Hz, resonance (internal magnification of the applied vibration) would probably damage it.

Most acoustic investigations are limited to the frequencies of human hearing, roughly 16–16,000 Hz. Microphones and vibration equipment show, of course, that sound and vibration extend outside those limits (see Figure 16).

Mass? Weight?
One final bit of math. Force (F), according to Newton, equals mass (M) × acceleration (A). If M happens to be 10 kg and if peak acceleration A happened to be 501 m/s², then F = 5010 newtons (N). That’s how the International System works. U.S. engineers do this calculation differently, but they get essentially the same answer. They say F = weight (W) × A. Instead of 10 kg mass, they use 22 lb. weight and multiply this by 51.1 g, so that F = 1124 lb. force, sometimes indicated lbf.

Summary
Vibration is one of the most widely measured properties. Of the myriad forms of vibratory motion, sinusoidal vibration is the simplest and most easily grasped. Although never found outside test labs and the theoretical realm, single-frequency-at-a-time sinusoidal vibration is useful as a point of entry into the dynamic behavior of the real world.

This article was adapted from Chapter 2 of the forthcoming textbook, A Minimal-Mathematical Introduction to the Fundamentals of Random Vibration and Shock Testing, HALT, ESS and HASS, also Measurement, Analysis and Calibration, by Wayne Tustin.