Plotting and Graphing

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1 Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured dt. Properly presented, grphs provide compct nd concise delivery of informtion. However, poorly prepred grphs cn be confusing nd misleding. There re severl computer pckges vilble for producing grphs, but mny of these do not generte dequte engineering style grphs without extensive modifiction to the defult prmeters. Remember tht computers re simply tools, never use the excuse tht's wht the computer gve me for poorly prepred grph. The rules given below re for grphs with liner scles on both the horizontl nd verticl xes. Rules for other scles re similr nd will be presented in lter section. Rules for Grphing ) Grphs re lmost lwys prepred by plcing the dependent (output) vlue(s) long the ordinte (verticl xis) nd the independent vrible (input) long the bsciss (horizontl xis). 2) Scles for both xes should be selected for redbility nd clrity. Unless there re unusul circumstnces, select scles tht re multiples of, 2, 5,, 2, 5, etc. These scle vlues mke the grph much esier to use nd llow esier interpoltion between dt vlues. 3) It is usully desirble to show zero t the strt of both the ordinte nd bsciss unless this would compress the grph significntly. This rule is often difficult to pply, but use common sense ("engineering judgment") to determine if zero should be on both xes. 4) Grph pper should either be of the pre-printed type, crefully prepred blnk pper, or computer-generted. If hnd-drwn, xes should be drk nd thick lines drwn with stright edge for netness nd clrity. Mrkers should be plced regulrly long ech xis t the given scle divisions. If grid lines re used, they should be fint nd unobtrusive. 5) Ech xis should be indented from the edge of the grph pper to llow dequte lettering of the xis. Normlly the bsciss is long the bottom of grph nd the ordinte long the left hnd edge. If the grph requires longer bsciss, the grph should be plced with the bsciss opposite the bound side of the grph nd the ordinte long the left hnd side. 6) Ech xis of the grph must be lbeled long with the units used. The xis lettering is usully plced outside the xis nd its scle mrkers. Absciss lettering is oriented normlly nd ordinte lettering is oriented such tht it cn be red if the pper is rotted 9 degrees clockwise. 7) Grphs of theoreticl reltionships (given by explicit formuls) do not hve individul point designtions, either solid, dshed, or dotted lines re used. Mesured dt re presented with individul symbols plced t ech dt point. Smooth curves or stright line segments re sometimes used to connect the dt point symbols. Symbols cn include filled nd open circles, tringles, boxes, dimonds, etc., depending on how mny different types of ordinte dt re plotted. Empiricl reltionships re often plotted with symbols plced regulrly long the bsciss, with smooth curves of the ssumed type connecting the dt points.

2 8) If more thn one set of dt is plotted, legend is provided to identify ech set of dt. The legend is typiclly plced on the grph (wy from ny dt points), below the bsciss xis, or to the right of the grph. 9) Most grphs should be titled, with the dependent vrible nme given first. For exmple, grph entitled "Wter Density vs. Temperture" would indicte tht wter density ws the dependent (ordinte) vrible nd temperture ws the independent (bsciss) vlue. The title should be plced on the grph where it will not interfere with the other informtion given. Note tht in report the figure title will be prt of the cption, which is locted immeditely below the grph. ) On plot prepred by hnd (i.e. not computer generted) the nme of the grph preprer nd the dte re often plced in the lower right hnd corner of the grph pper (not inside the grph xis boundries). Log nd Semi-log Plots In mny sets of engineering or scientific dt, input nd/or output vlues re spred over n extremely wide rnge. For exmple, the dt given on the right represents the idelized mgnitude response of simple low-pss filter. Plotting this dt using liner bsciss would compress the dt gretly nd mke it dificult to see the rther shrp trnsition between nd 3 rd/sec. This type of dt (with the bsciss vlues spred over severl orders of mgnitude) would normlly be plotted on semi-log grph. On semi-log grph one of the xes (usully the bsciss) hs logrithmic scle. In engineering, this log scle is usully log bse, lthough ny other log scle is possible. A quick review of logrithms is pproprite t this time. The definition of logrithm cn be developed from simple eqution: x = N where is ny positive number except nd N is ny positive number. Another eqution cn be defined from this first eqution, x = log ( N) which sttes tht x is the logrithm bse of the number N. A few simple exmples using bse re: log ( ) = log ( ) = 2 log ( ) = 3 since =, 2 =, nd 3 =. Some of the commonly used properties of logrithms re given below: Nturl Frequency Mgnitude (rd/sec) rtio

3 log ( MN) = log ( M) + log ( N) M log = log ( M) log ( N) N log ( M ) = n log ( M) n Mny times the ``nturl" logrithm is used, nd is commonly denoted by ln insted of log. Nturl logrithms use the irrtionl number e s bse, where e is defined by e = lim( + x) / x Logrithms to ny bse cn be defined using the nturl logrithm by the reltionship log ( N) = ln ( N) / ln ( ) x The originl dt tble is given t the right with the ddition of column contining the log (log bse ) of the nturl frequency. This set of dt vries over much smller rnge of dt vlues, nd cn be plotted on liner scles s shown in Figure below..2 Nturl Freq, ω log ω Mgnitude rd/sec rtio Mgnitude Rtio log (ω) Figure. Mgnitude rtio vs. log(ω).

4 A much simpler lterntive to tking logrithms, then plotting on liner scles is to simply use logrithmic scles. Preprinted semi-log ( log nd liner scle) nd log-log (two log scles) grph pper is redily vilble in vriety of ptterns. Mny softwre pckges (such s Excel) will generte plots directly with logrithmic xis on either the ordinte or bsciss (or both). The sme dt of mgnitude rtio vs. frequency ω is plotted on logrithmic xes in Figure 2..2 Mgnitude Rtio Frequency, ω (rd/sec) Figure 2. Mgnitude rtio vs. frequency, ω (log xis). To crete logrithmic xis in Excel, double click the xis, then check the logrithmic scle box within Formt Axis. The minor gridlines must be turned on by checking the pproprite box in Chrt Options.

5 Linerizing nd Grphing Dt We hve ll used the fmilir stright line s n pproximtion to set of dt. In your erly chemistry nd physics lbs you lso pproxinmted dt with other types of curves, such s exponentil or logrithmic. Another clss of problems cn be solved using trnsformtions to modify dt such tht the stright line pproximtion will work gin. A set of dt (Speed vs. Time) is plotted below in Figure 3. There does not pper to be good fit between the dt points nd ny stright line. Speed (m/sec) Time (sec) Figure 3. Speed vs. Time. However, if we look t the dt crefully, we might note tht reltively simple trnsformtion will llow us to plot stright line. We form column of /T s well s T itself nd plot Speed vs (/Time) in Figure 4. Note tht stright line pproximtion is much more resonble with this trnsformed dt. Speed (m/sec) /Time (/sec) Figure 4. Speed vs. (/Time).

6 A tble of commonly encountered trnsformtions is given below. Item () is simply the common stright line formul. The trnsformtion used in the exmple bove is given in item (3). Items (5) nd (6) represent dt tht plots s stright line on semi-log pper. Item (7) is represents dt tht plots s stright line on log-log pper. Tble Stright Line Trnsformtions* Note: Y = A + B X y = f(x) Y X Intercept, A Slope, B ) y = + bx y x b 2) y = + b x y x b 3) y = + b/x y /x b 4) x x/y x b y = + bx 5) y = b x log(y) x log() log(b) 6) y = c bx log(y) x log() b log(c) 7) y = x b log(y) log(x) log() b 8) y = + bx n y x n b (*from Experimentl Methods for Engineers, J.P. Holmn, McGrw-Hill, 989)

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