Chaper Engineering Soluions.4 and.5 Problem Presenaion Organize your work as follows (see book): Problem Saemen Theory and Assumpions Soluion Verificaion Tools: Pencil and Paper See Fig.. in Book or use Analysis Sofware, e.g. Mahcad Tools: Word Processor See Fig..a and b in Book Benefis: Neaer appearance Impor graphics Impor resuls from oher ools, such as spread shees Source: Fig..a
Analysis Sofware : Advanages: Always clean and organized Numerics will be correc (assuming you enered correc equaions) Auomaed graphing and presenaion ools Superior error and plausibiliy checking Analysis Sofware : So why aren you using Mah sofware ye? Examples of Analysis Sofware: Mahemaica (symbolic) Maple (symbolic) Mahcad Malab (numerical) Numerous specialy producs (general and symbolic) Dipole (physics) analysis example Mahemaica Mahemaica Mahemaica
Mahcad Calculus Example x yx () + x + x sin( x) Maple Differeniaion Symbolics Example (Mahcad) Examples in Mahcad: compue moion of sliding block. d dx yx () wih a x ( ) + x + x sin( x) ( ) x + x sin( x) x + x + x sin( x) ( ) ( ) + x + x sin( x) cos( x) a Moion of sliding block. Dynamic sysem analysis example Mahcad can find he soluion by symbolic Equaion solving: Dynamic sysem analysis example Dynamic sysem analysis example
Wha is in i for me? Yes, you will have o ge used o he consrains imposed by he sofware. This will pass. All learning is an invesmen for your fuure. Wha is in i for me? Benefis: You will be Faser More Efficien More accurae. Beer presenaion Time is money. Wha is in i for me? Tools such as Mahcad allow you o creae: Beer presenaions Accurae resuls. Beer design choices (play wha if? scenarios) Conclusion Chaper Plan for he long erm. Become familiar wih hose ools ha will make you he mos producive. Your invesmen will pay off handsomely. Chaper 4 Represenaion of Technical Informaion A Typical Scenario We colleced daa in an experimen. The daa se migh consis of a lis, such as he one on page 4 in your book, or a compuer daa file. We plo he daa. 4
A Problem: Noisy Daa (Noise ofen resuls from poor qualiy measuremens, or from inerference (jus ry AM radio) How good is his conrol? DC Servomoor: Mahemaical Model and Validaion Anoher Example: Conrolling a DC Servomoor Engineers mus Collec Informaion (Daa) Creae Records Analyze and display he informaion (e.g idenify rends, creae a mahemaical model A se of daa 5
An Example: A sored se of daa from Tensile Tesing of Maerials A Tensile Tesing Machine Maerial samples are insered and he force o break he sample apar is recorded. Force Force Firs Column: Force (in Kilo- Newons) required o break he sample Force Number 5 57 5 57 5 57 54 547 55 557 5 57 57 577 58 587 Second Column: Number of samples broken a he respecive Force Level Toal 5 Samples Frequency.5.5.5.5 frequency 4 8 4 Series Series Cumu 5 57 5 57 5 57 54 547 55 557 5 57 57 577 58 587 Force in KiloNewons We can ener he daa se ino a spreadshee program such as MS Excel, and plo he informaion in various formas. 5 5 5 5 54 55 5 57 58 59 - value Ploing in various formas: Same daa, Line graph (in blue) Cumulaive (adding all samples) in red
frequency 4 8 4-557 57 5 5 57 5 57 57 5 57 54 54 547 55 547 55 557 5 57 57 557 5 57 57 577 577 58 587 58 587 5 5 5 5 54 55 5 57 58 59 value Series Series Cumu You have various opions o highligh daa. Explore hem and find ou wha works bes. frequency 4 8 4-5 5 5 5 54 55 5 57 58 59 value For your Homework assignmen: Please use spreadshee sofware or Mahcad! Explore he bes opions o presen he informaion, and submi Series Series Cumu Mahcad Example: A Gaussian (Normal) Disribuion. The numbers are shown a righ. f = 4 5 7 8 9 4 5 5 79 97 4 8 74 7 4 f 5 4 5 8 4 4 8 µ 4 σ Hisogram Ploing he Gaussian (Normal) Disribuion. (Hisogram) in µ + 4 σ 5 4 f F( in). 5 8 4 4 8 µ 4 σ Hisogram Normal fi Compue he Normal Disribuion. (Blue Line) in µ + 4 σ Mahcad Commands: Hisogram: f Gaussian Fiing Funcion: := his( in, N) ( ) Fx ():= nh dnorm x, µ, σ For help in Mahcad, see Quick shees Saisics 7
Chaper 4. Collecing Daa Manual (slow, inefficien, error-prone. don wase your ime! Someimes, of course, manual recording of daa is expedien) Compuer assised (ypically faser and more accurae) You can also buy special recorders (daa loggers) ha record very large quaniies a very high raes. Example: During Nuclear esing a he Nevada Tes Sie, all daa mus be colleced wihin abou nanoseconds afer riggering. The insrumenaion is desroyed by he explosion Ploing Experimenal Daa: A se of x/y daa x 4 5 7 8 9 = yx () = 9.87.9 5.74 7.4.8.7 7.88 8.495.5 4.55 Ploing Experimenal Daa: Basics Presen he informaion clearly and concisely! Each graph should speak for iself: Label he axes! Descripive Tile! Page 55 Fig. 4.9 Scaling he Axes Page 55 Fig. 4. Please Read and apply! Axes Graduaions 8
Page 57 Fig. 4. Page 57 Fig. 4. Proper Represenaion of Daa You choose. Ploing Experimenal Daa: Graphing 4 he daa (scaer poins) 5 Ploing Experimenal Daa: Graphing he daa (daa poins 45 4.5 conneced 4 by lines) 5 Y 5 Y 5 5 5 4.5 4 5 7 8 9 X 5 4 5 7 8 9 X We can use Bar Graphs 45 4.5 4 5 45 4.5 4 5..Or seps Y 5 Y 5 5 5 4.5 4 5 7 8 9 X 4.5 4 5 7 8 9 X 9
Linear Inerpolaion: 45 4.5 Bes fi line 4 5 Page Fig. 4. Y µ x+ β 4.5 5 5 4 5 7 8 9 Xx, Muliple Daa Ses Page Fig. 4. Spreadshee Rules Ploing Experimenal Daa: A Quadraic Funcion (free fall) The falling disance is proporional o ime :=,.. := g Free Fall: Elev. vs. Time Drop in Elevaion (meers) 5 49.5 4. Free 49.5 Fall: Elev. vs. Time Same daa, in log-log forma Drop in Elevaion..49 4 8. Time in Seconds.49... Time in Seconds
Drop in Elevaion (meers). 49.5. Log-Log. plos: Wha is differen? 49.5 The axis labels are muliples of, No incremens by, as in linear graphs Drop in Elevaion (meers)..49... Time in Seconds.49... Time in Seconds Page 8 Page 9 Log-Log plos: Wha is differen? The axis labels are muliples of, In he linear graph, 5 he daa were evenly 49.5 disribued. Drop in Elevaion (meers) 4 Log-Log plos: Wha is differen? The axis labels are muliples of,. 49.5 In he logarihmic plo, he daa seem o be clusered in he upper righ corner. Why? Drop in Elevaion (meers)..49 4 8.49... Time in Seconds. Time in Seconds
d :=.,... f( d) := log() d d = f( d) =. -.4 -.98.7 -.55..4..4.9.79..4.5.98.8.447..49.4.5 7 58 fd ( ).5.5 4 8. d The (decadic) logarihm of. = -. Log()= ; Log() = We can use logarihmic plos o es a daa se for polynomial relaionships. Look a hese hree polynomials: f( x) := x.5 f( x) := x f4( x) :=. x.5 Now graph he hree polynomials in loglog forma:. 4.795 f() x f() x f4() x f ( x) := x.5 f ( x) := x f4 ( x). x.5.. := We can use loglog graphing o idenify paerns. Example: Tesing he daa Se a righ for Polynomial Properies. x.4.8...4.8 4. 4. 5 5.4 5.8.. 7 = fp( x) =.85.4 7.48 94.9. 9.97 98. 54.74.5 8...5.5.8..8 Here is a linear plo of he daa. The values are somewha fp( x) scaered due o sensor noise. 8 7 5 4 4 8 x
Here is a loglog plo of he same daa.. 4 7.7 The values fp( x) appear o follow a sraigh pah.. A bes fi line is found as: fp x. 4 7.5 fpx () fpx (). ( ) := 7 x 5.4 x 9.8 7 x 9.8