Page 22 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 1.2 Periodic Motion

Figure 1.1-5. Vector z and its con­
jugate z*.

Figure 1.1-5 shows z and its conjugate z* = which is rotating in the
negative direction with angular speed -a;. It is evident from this diagram, that the
real component x is expressible in terms of z and z* by the equation
x = ^(z + z*) = A cos ù)t = Re Ae"' (1.1-9)

where Re stands for the real part of the quantity z. We will find that the
exponential form of the harmonic motion often offers mathematical advantages
over the trigonometric form.
Some of the rules of exponential operations between Zj = A^e^^^ and Z2 =

Multiplication 2,22

Division ( 1.1-10)
^2 1^2 ;

Powers
^\/n _ j^\/n^ie/n

1.2 PERIODIC MOTION

It is quite common for vibrations of several different frequencies to exist simulta­
neously. For example, the vibration of a violin string is composed of the fundamen­
tal frequency / and all its harmonics, 2/, 3/, and so forth. Another example is the
free vibration of a multidegree-of-freedom system, to which the vibrations at each
natural frequency contribute. Such vibrations result in a complex waveform, which
is repeated periodically as shown in Fig. 1.2-1.
The French mathematician J. Fourier (1768-1830) showed that any periodic
motion can be represented by a series of sines and cosines that are harmonically
related. If x{t) is a periodic function of the period r, it is represented by the