Introductory Mathematics for Engineering Applications

Sinusoids in Engineering

CHAPTER 6

A sinusoid is a signal that describes a smooth repetitive motion of an object that oscillates at a constant rate (frequency) about an equilibrium point. The sinusoid has the form of a sine (sin) or a cosine (cos) function (discussed in Chapter 3) and has applications in all engineering disciplines. These functions are the most important signals because all other signals can be constructed from sine and cosine signals. A few examples of a sinusoid are the motion of a one-link planar robot rotating at a constant rate, the oscillation of an undamped spring-mass system, and the voltage waveform of an electric power source. For example, the frequency of the voltage waveform associated with electrical power in North America is 60 cycles per second (Hz), whereas in many other parts of the world this frequency is 50 Hz. In this chapter, the example of a one-link robot rotating at a constant rate will be used to develop the general form of a sinusoid and explain its amplitude, frequency (both linear and angular), phase angle, and phase shift. The sum of sinusoids of the same frequency will also be explained in the context of both electrical and mechanical systems.

6.1 ONE-LINK PLANAR ROBOT AS A SINUSOID

A one-link planar robot of length l and angle θ is shown in Fig. 6.1. It was shown in Chapter 3 that the tip of the robot has coordinates x = l cos θ and y = l sin θ. Varying θ from 0 to 2π radians and assuming l = 1 (l has units of x and y), the ...