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EGR 1101 Unit 6

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 .

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EGR 1101 Unit 6

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  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 • 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 • One-link robot in motion • Simple harmonic motion of a spring-mass system • Adding sinusoids in an RL circuit

  6. One Question, Three Answers • Three equivalent answers to the question, “How fast is the robot arm spinning?” • Period, T, unit = seconds (s) • Tells how many seconds for one revolution • Frequency, f, unit = hertz (Hz) • Tells how many revolutions per second • 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 cosy + cosx sin y sin(x y) = sin x cosycosx sin y and cos(x + y) = cosx cosy sin x sin y cos(x y) = cosx cosy+ sin x sin y

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