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                              How It’s Used

13.3 Natural Frequency

The following sections present some examples of how imaginary numbers and the complex plane are used in the fields of aeronautical and electronic engineering.  Both discussions require a fair amount of new material to get even a general understanding.

As we saw in Chapter 12, the fields of mechanical vibrations or alternating currents use very similar mathematics.  A key concept that is used in both fields is natural frequency.

All flexible objects have a characteristic known as a natural frequency. For example, each string on a guitar sounds different because its length and/or material cause it to have a different natural frequency.  The natural frequency, f_n, of an object is the frequency at which an external force will cause the maximum deflection.  An object’s natural frequency is related to its spring constant by the following equation:

                                                                                       (13-9) 

Figure 13.10 shows some equations that give the spring constants of different types of objects.

Designers of buildings in areas with earthquakes need to keep the natural frequencies of their buildings very different from the frequencies of the earthquake motions they expect.  If the building’s natural frequency is the same as the earthquake frequency, the building will experience the maximum deflection and will likely be damaged. The designers also need to avoid buildings that have sections with different natural frequencies. An L-shaped building has sections with different natural frequencies.  When an earthquake occurs, the sections will move very differently rather than together.  The different motions may damage the building.

Some engineers design to meet the natural frequency of an object.  A microwave oven is a machine that emits waves at the natural frequency of a water molecule.  This causes the water molecules to vibrate at the maximum deflection.  The water then gets very hot.

 13.4 Aircraft  Stability and Control

The concept and math of the spring/mass/damper problems also apply to the study of aircraft stability and control.  In the shock absorber example, the motion was linear, along the axis of the shock absorber.  Aircraft motion involves not only linear motion (airspeed, altitude) but also rotation (pitch, roll and yaw).  The tail of the aircraft acts like a weather vane to keep the nose of the aircraft pointing forward.  The tail's effect is similar to both the spring and the damper in the shock absorber problem.  The size of the tail and the distance behind the aircraft's center of gravity  (also considered the center of rotation) both affect the c and k values of the aircraft's equation of motion.  Like the shock absorber, the effect of the tail determines the oscillation and damping of the aircraft.  These equations allow designers to analyze the aircraft's motion due to changes in the direction of the wind or the pilot changing the direction of the plane.  Engineers do extensive testing on the aircraft's response to ensure it flies stable with the right 'feel' to the pilot.  If the aircraft is unstable or too poorly damped, the pilot will find it too difficult to control.  If the aircraft is too heavily damped, the pilot will consider the aircraft too sluggish and unresponsive.

The stability and controllability of an aircraft is analyzed using a set of differential equations.  One of these equations, simplified and transformed, takes the form: 

                                                                    (13-10) 

where     f_n = the natural frequency of the plane in radians/second.

                                d = the damping ratio (c/c_c where c_c = critical damping) 

 Both of these values are determined through wind tunnel and flight tests.

 An equation for a Lear Jet works out to:

                                                                      (13-11)

 If we use the quadratic formula, we get the following roots: 

s₁ = -0.88 + 1.29i 

s₂ =-0.88 - 1.29i 

Aeronautical engineers then take these roots and plot them on the imaginary plane.  The roots allow the engineers to know instantly how the plane will perform.  Figure 13.12 illustrates the stability/oscillation condition for each type of root.

If the roots are complex conjugates, the plane will oscillate.  If the root is real, the plane will not oscillate.

 If the real component of the root is negative, the plane will be stable (oscillate into control).

 If the real component of the root is positive, the plane will be unstable (oscillate out of control).

 If the root is a negative real number, the plane will be overdamped (not oscillate).

 If the root is a positive real number, the plane will be purely divergent (loose control without oscillating). 

As can be seen from Figure 13.12, the roots for the Lear Jet fall into the “Oscillatory but stable” range.

As designers modify the plane (e.g., change the tail size or move a fuel tank) they look at how the roots change or they track the root locus.  Analyses like this can be done for nearly any type of object that oscillates or vibrates.