Introduction to Engineering Linear & Logarithmic Scales Agenda Linear & Logarithmic Scales Linear vs. Linear Plot Log-Log Plot Semi-log (log v. linear) plot Display Data - 1

Working With Data Engineers often collect large amounts of data Two common ways to analyze the data are: Write an equation to fit the data. Graph the data. Graphs allow the engineer to spot trends in the data more easily than they can by looking at the numbers in a table. Graphs are also a good way to show other people the results of your work. Writing an equation that fits the data allows you to intrapolate or extrapolate. Display Data - 1

Working with Equations Describing Engineering Phenomena and Data Plotting data with appropriate scales can be very helpful in display of data and interpretation Choosing the right scale can make the appear on a straight line Choosing the right scale allows you to plot data ranging over several orders of magnitude on a single page. Suggested References: Croft, Meyers, Boyer,Miller, And Demel. 1989. Engineering Graphics, McGraw Hill. P. 541- 558. Baumeister and Marks. Standard Handbook for Engineers. The idea of this section is to give students a very brief start on thinking about data and how it might be described with equations and how it might be graphed. They will be collecting data in lab in the near future. This will introduce plots or ways of linearizing three different types of expressions commonly found in engineering problems. Display Data - 1

Linear and Logarithmic Scaling -20 -15 -10 -5 5 10 15 20 Linear axis: Logarithmic axis: 0.1 1 10 100 Display Data - 1

Three Examples Temperature Expansion Wind Force Population Growth Linear Y = mX + B Wind Force Power Equation Y = bXm Population Growth Y = bemX A physical example of each type of equation will be given. For the second two, it is shown how the equation can be modified to have a linear (or straight line) form by using semi-log or log-log graph paper. It is easier to interpolate or extrapolate if the data can be graphed in a straight line. Display Data - 1

Linear Relationship Example Temperature Expansion of Pipe Length, L L = a (T – To) + Lo L = a (dT) + Lo (Form : Y = mX + b) dT L Lo a 1 In physics students may have done calculations on the changes in dimensions of an object due to temperature changes. The change in length is generally directly proportional to change in temperature. This implies a linear relationship. Note for students that T- To means change in temperature from a reference value (To), also shown as delta T (dT) in second line. Thermal expansion of building or structural components is a good example of a linear relationship. The handrails on bridges and PVC electrical conduit often have slip joints to accommodate the changes so that damage is not done by the expansion and the integrity of the structure is protected. If there were no provision for expansion, the structure would buckle. Display Data - 1

Power Relationship Example Wind Force on Road Sign Wind Force on a Sign F = P A P = 0.0036 V2 F = 0.0036 V2 A Where P = pressure in lb/ft2 V = velocity in ft/sec (Form Y = bXm) The idea here is to think about how heavy or how strong the frame or posts must be to support the road sign. From F = PA it can be pointed out that for a given pressure, the force is directly proportional to area. However the Pressure equation shows that the pressure goes up as the square of the velocity. (Basic power equation form) Help student see how p maps to Y, 0.0036 A to b, V to X and 2 to m Display Data - 1

Y = bXm The first plot just shows how the equation would look if plotted on linear x and linear y axis. Note that the equation is curved. Curvature depends on m. Note: Awareness is what we want in next three slides. Students may not be familiar with log and ln. May not have had it in math class yet. Hopefully this will alert them to the fact that this is useful when they do see it in math class. Display Data - 1

Y = bXm Linear Form: log Y = log b + m log X Y* = m X* + b We can see from the second expression if we take the log of both sides of the equation we get a new equation where if we substitute log Y for Y and log X for X we will get a linearized (straight line equation). Note if we plot the log of X vs log of Y we get a straight line, the second plot. Display Data - 1

Y = bXm Linear Form: log Y = log b + m log X Y* = m X* + b The more common way would be to plot on a log vs. log scale, using log vs log graph paper or log function in spread sheet. Note that this is a log scale, not a linear scale. If we had been plotting data, and we got this straight line on the log vs. log plot, we would know that the basic relationship between the X (independent variable) and y (dependent variable) is a power equation form. From the slope of the plot we can get m, from the intercept we can get b. Also note that the log scales allow for a wide range of data, since each segment represents a power of 10. Display Data - 1

Exponential Relationship Example Population Growth Y = bemX Linear Form: ln Y = ln b + m X Y* = m X + b Population growth, like microbial growth in food would follow this form of equation. Here linearizing the equation, by taking ln (natural log) of both sides shows that if we plot the Y axis as logarithmic and the x as linear, we will get a straight line for this data. Note that pressure loss in a pipe as a function of velocity also follows this form of equation. Display Data - 1

Assignment Assignment #8 Log and Semi-Log Review sheet Bring a floppy disk to class tomorrow