1. EGR 1101 Unit 6 Sinusoids in Engineering (Chapter 6 of Rattan/Klingbeil text)

2. 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: http://en.wikipedia.org/wiki/Sinusoid http://en.wikipedia.org/wiki/Sinusoid  Sinusoids are the most important of these.

3. 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

4. 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

5. 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

6. 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

7. 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  = 2f = 2/T

8. 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.

9. 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.

10. 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?

11. 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