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12.3 Sinusoidal Functions

In the sections before, we learned how the trigonometric functions can be used to calculate distances.  By looking at their graphs, you can see how the trigonometric functions are cyclical.  For all of the functions, each rotation around the unit circle creates new cycle.

The sine and cosine functions are used to model most cyclical motions.  This section discusses some sinusoidal relationships and two of their key parameters, amplitude and frequency .  More sophisticated methods of modeling cyclical motions are discussed in courses teaching Fourier analysis.

12.3.1 Amplitude 

By using the cosine function we can expand on the one-arm curl problem discussed in the Weight Room Mechanics chapter.  A close look shows that there is a sinusoidal relationship between  and the moment that is created by the weight.  While calculating the moment which resulted, we were limited by the need to know the perpendicular distance, l, to the line of force of the weight, w.  Recall that: 

M = Fl     where:

                                                            M = the moment

                                                            F = the force, and

                                                            l = the length of the moment arm.

With the cosine function, we can calculate the perpendicular distance, and therefore the moment, for any angle .  In this example, the length of the moment arm, l, equals the length of the adjacent leg, a.

From the diagram, we can see that:

The moment M equals Fl, where l = a = 14 in. and F = w = 10 lb. Therefore the moment at any angle  is given by the equation:

M  = Fl = (14)(10)= 140cos ,  with units in in.lb.

An engineer who comes across the same configuration in a piece of machinery may need to know the exact moment for every value of .  For this configuration the function of the moment would be graphed as follows:

Although the one-arm curl doesn't involve a full -cycle, the problem represents an example of the concept of amplitude.  Sinusoidal functions involve the oscillation between maximum and minimum values.  In this example, the magnitude of the moment fluctuates as a function of .  The moment will be at its highest when  = 0.  This is when the forearm is fully extended and the moment arm is the longest.  When the forearm has rotated  ( /2 radians), the length of the moment arm is 0 and there is no moment.  The moment will get no higher than 140 in. lb.  The value of 140 is therefore the amplitude of the function M = 140 cos .

The amplitude represents the maximum value of a sinusoidal function.  Recall that the sine and cosine functions fluctuate between 1 and -1.  When a coefficient is placed before the function, the amplitude of the sine wave is affected.  The graphs of y = 3sin , y = sin , and y = 1/3sin are shown in Figure 12.22.

Note that the amplitude of 3sin is three times that of sin .  The amplitude of (1/3)sin is one third of sin .  Also note that the three functions all cross theq-axis at the same locations.

12.3.2 Frequency 

The frequency of a cyclical function is one of its most important characteristics. In the example of an athlete jumping rope, we used the sine function to determine the height of the tip of the rope as a function of the angle of the rope.  When working with rotating parts of machinery, designers are usually more interested in the position as a function of time. To get position as a function of time, we would need an equation of the form:

                                                y = sint             where t = time, in seconds

However the trigonometric functions can only measure angles.  To get the sine as a function of time, we need a coefficient in front of the t.  The coefficient which is usually used in cyclical (or harmonic) motion is .  The product of the t term must have units of degrees or radians.  For t to be in radians,  must have units of radians per unit time.  This of course is the standard unit of angular velocity.  In trigonometry there is a direct relationship between frequency and angular velocity .

Quite often, the frequency of a motion is defined in terms of cycles per unit time. When the frequency is given in terms of cycles per second, it is usually denoted with the variable f. A common unit of frequency is the hertz (Hz).  A hertz represents one cycle per second.  The megahertz (MHz) used by radio stations to identify their frequencies represents a million cycles per second.  The relationship between frequency, f, and angular velocity, , is given by the equation:

w = 2p f

To be able to model cyclical motions with the trigonometric functions, one needs to be able to convert frequencies given in cycles per second to frequencies of radians per second.  If an athlete jumps rope at 2 revolutions per second, the angular velocity is:

The equation of                where   t = time, in seconds

                                                                        y = the height of the rope, in meters

            gives us the height of the rope relative to his hands at any time, t.

Figure 12.23 shows sine waves having different frequencies:

Note that the sinx function fits in three cycles by the time the sin(x/3) function has completed one cycle.  The time it takes a trigonometric function to complete one cycle is known as its period and is usually denoted by the variable T. The period of a function is the inverse of its frequency.  In mathematical terms:

               (12-14)

The period of the sinx function is 2 .  Its frequency is therefore 1/2 .  The period of the sin(x/3) is three times the length of the period of the function sinxor 6 .  If the period is three times that of sinx, its frequency is one-third of sinx.  Note that the amplitudes of the functions remain the same when only the frequencies are affected.

The significance of different frequencies can be further explained by the example of hurdling.  Hurdling could be viewed to involve two motions, running and the jumping of hurdles.  The running strides are made at a particular frequency while the hurdling is done at a different frequency.  A hurdler who 3-steps will make three strides with each leg between each hurdle.  If a hurdler 3-steps with a stride at a frequency of x, the hurdling will have a frequency of (1/3)x.