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Applications of Derivatives in Electrical Circuits and Beam Mechanics

This lecture covers the application of derivatives in electrical circuits and beam mechanics, focusing on key concepts such as voltage, current, power, and how they relate to inductors and capacitors. It introduces the product and chain rules for derivatives, provides graphical interpretations, and explores beam terminology and loads including simply supported and cantilever beams. The importance of flexural rigidity and how geometry and material affect beam behavior are discussed, along with examples of deflection and stress in beams.

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Applications of Derivatives in Electrical Circuits and Beam Mechanics

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  1. EGR 1101: Unit 9Lecture #1 Applications of Derivatives: Electric Circuits (Section 8.4 of Rattan/Klingbeil text)

  2. Review: Some Derivative Rules where a, c, n, and  are constants.

  3. Two New Derivative Rules • (Product rule) • If f(g) is a function of g and g(t) is a function of t, (Chain rule)

  4. Today’s Examples • Voltage, current, & power • Current & voltage in an inductor • Current & voltage in an inductor (graphical and working backwards) • Current & voltage in a capacitor • Current & voltage in a capacitor (graphical and working backwards)

  5. Review: Some Electrical Quantities

  6. Review: More Electrical Quantities

  7. Voltage-versus-Current Relations • For resistors, • For inductors, • For capacitors,

  8. EGR 1101: Unit 9Lecture #2 Applications of Derivatives: Beams (Section 8.5 of Rattan/Klingbeil text)

  9. Some Beam Terminology • Types of beams • Simply supported • Cantilever • Types of load on a beam • Concentrated • Distributed

  10. More Beam Terminology • In addition to the type of beam and load, a beam’s behavior also depends on its geometry and the material it is made of. • Its geometry is summarized in a quantity called the second moment of area (I). • Its material is summarized in a quantity called the modulus of elasticity (E). • The product of these two (EI) is called the flexural rigidity.

  11. Some Beam Quantities

  12. Excellent Online Resource • University of Wisconsin’s online lessons on Strength of Materials: http://www3.uwstout.edu/faculty/scotta/upload/Foley-StaticsStrengths.pdf • See especially Topic 4 (Beams) and Topic 8.2 (Stress on Incline Planes).

  13. Review • Given a function f(x), the function’s local maxima occur at values of x where and • Its local minima occur at values of x where and

  14. Today’s Examples • Deflection of a cantilever beam with end load • Deflection of a simply supported beam with central load • Deflection of a simply supported beam with distributed load • Maximum stress under axial loading

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