EGR 1101 Unit 6 Sinusoids in Engineering (Chapter 6 of Rattan/Klingbeil text)
Periodic Waveforms Often the graph of a physical quantity (such as position, velocity, voltage, current, etc.) versus time repeats itself. We call this a periodic waveform. Common shapes for periodic waveforms include: Square Triangle Sawtooth Sinusoidal See diagram at bottom of page: Sinusoids are the most important of these.
Sinusoids A sinusoid is a sine wave or a cosine wave or any wave with the same shape, shifted to the left or right. Sinusoids arise in many areas of engineering and science. We’ll look at three areas: Circular motion Simple harmonic motion Alternating current
Amplitude, Frequency, Phase Angle Any two sinusoids must have the same shape, but can vary in three ways: Amplitude (height) Frequency (how fast the values change) Phase angle (how far shifted to the left or right) We’ll use mathematical expressions for sinusoids that specify these three parameters. Example: v(t) = 20 sin(180t + 30 ) V
Today’s Examples 1. One-link robot in motion 2. Simple harmonic motion of a spring-mass system 3. Adding sinusoids in an RL circuit
One Question, Three Answers Three equivalent answers to the question, “How fast is the robot arm spinning?” 1. Period, T, unit = seconds (s) Tells how many seconds for one revolution 2. Frequency, f, unit = hertz (Hz) Tells how many revolutions per second 3. Angular frequency, , unit = rad/s Tells size of angle covered per second
Relating T, f, and If you know any one of these three (period, frequency, angular frequency), you can easily compute the other two. T = 1/f = 2f = 2/T
General Form of a Sinusoid The general form of a sinusoid is v(t) = A sin (t + ) where A is the amplitude, is the angular frequency, and is the phase angle. Often is given in degrees; you must convert it to radians for calculations.
Adding Sinusoids Many problems require us to find the sum of two or more sinusoids. A unique property of sinusoids: the sum of sinusoids of the same frequency is always another sinusoid of that frequency. You can’t make the same statement for triangle waves, square waves, sawtooth waves, or other waveshapes.
Adding Sinusoids (Continued) For example, if we add 10 sin (200t + 30) and 12 sin (200t + 45) we’ll get another sinusoid of the same angular frequency, 200 rad/s. But how do we figure out the amplitude and phase angle of the resulting sinusoid?
Adding Sinusoids (Continued) Our technique for adding sinusoids relies heavily on these trig identities: sin(x + y) = sin x cos y + cos x sin y sin(x y) = sin x cos y cos x sin y and cos(x + y) = cos x cos y sin x sin y cos(x y) = cos x cos y + sin x sin y