Root Mean Square Deviation Problem: To measure how hilly a landscape is or how rough a

Chapter 10, Problem 20

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Root Mean Square Deviation Problem: To measure how hilly a landscape is or how rough a machined surface is, people ask the question On average, how far do points on the surface deviate from the mean level? If you simply average the deviations, you will get zero. As Figure 10-3j shows, there is just as much area above the mean as there is below. One way to overcome this difficulty is to Square the deviations. Find the average of the squares. Take the square root of the average to get an answer with the same dimensions as the original deviations. Figure 10-3j The result is called the root mean square deviation. For instance, the roughness of a machined surface might be reported as 0.1 microinch, rms, where rms stands for root mean square and a microinch is one-millionth of an inch. a. Suppose that the deviations from average are sinusoidal, as in Figure 10-3j. That is, d = k sin x, where d is deviation, x is displacement along the surface, and k is a constant amplitude. Find the average of d2 for one complete cycle. Use the result to calculate the rms deviation. b. Plot the graph of y = sin2 x. Sketch the result. Show that the resulting graph is itself a sinusoid, and find its equation. c. Show that you can determine the answer to part a graphically from part b, without having to use calculus. d. Suppose a surface is lumpy, as in Figure 10-3k, and has the shape of the graph y = |sin x|. Find yav, the average value of y. Then find the rms deviation, using the fact that Deviation = y yav Based on your answer, would this surface be rougher or smoother than a sinusoidalsurface with the same maximum distancebetween high points and low points, as inFigure 10-3k?