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Here are some engineering simulations and artwork along with some Mathematica code. See also my Physics page.
CFM56-5 Turbofan Jet Engine - drawn in AutoCAD 2000, AutoLisp, rendered in POV-Ray 3.6.1, 3/2/07
I saw this cool looking poster of a turbofan engine schematic and it inspired me to make a 3D rendering of it. This engine is used for the Airbus. The streamlines shown here are just for visual effect. In reality, the streamlines would be turbulent. The blue streamlines show where cold air enters and bypasses the engine (turbofans typically have a high bypass ratio). Green streamlines are shown through the compressor stages. The red streamlines show where hot gas exits the burners and turbine stage. |
This was rendered from a turbofan blade model that was studied in our CFD lab.Links GE90-115B Gas Turbine - world’s most powerful jet engine EngineSim - Java applet for learning about jet engines, here is another one M400 Skycar - flying car, Vertical Take-Off and Landing (VTOL), be sure to see the interview with Dr. Moller ThrustSSC - first supersonic land speed record Miniature Harrier - amazing remote control AVAB Harrier with Pegasus engine by Ewald Schuster Jet-Powered Go Kart - 100 mph small jet engines - homemade, engine, model, red hot afterburner test remote control jet - 200 mph, here’s another one that goes 270 mph Chrysler Turbine Car |
3-Phase Brushless DC (BLDC) Motor - POV-Ray 3.6.1, 9/31/06
Brushless motors are popular in model airplanes. This one has a 6 pole stator and an 8 pole rotor. The technique used here is very similar to that used in Maglev trains. See also my Tesla coil magnetic field.Links Brushless Motor Schematics - by Gertjan Kool Finite Element Method Magnetics - free program by David Meeker |
Finite Element Method (FEM) Solution to Poisson’s equation on Triangular Mesh
solved in Mathematica 4.2, 5/24/06; mesh generated with Gmsh; old version: Matlab and DistMesh
 Here is some code to solve Poisson’s equation on an unstructured grid of triangular elements using the Finite Element Method (FEM): (* runtime: 0.1 second *) n = 10; nodes = Flatten[Table[{x, y}, {y, -1, 1, 2.0/(n - 1)}, {x, -1, 1, 2.0/(n - 1)}], 1]; elements = Flatten[Table[{{i, i + 1, i + n}, {i + 1, i + n + 1, i + n}} + (j - 1) n, {j, 1, n - 1}, {i, 1, n - 1}], 2]; nnode = Length[nodes]; nelem = Length[elements]; fixed = Map[Position[nodes, #][[1, 1]] &, Select[nodes, Max[Abs[#]] == 1 &]]; Kglobal = Table[0, {nnode}, {nnode}]; F = phi = Table[0, {nnode}]; Scan[(xy =Join[Transpose[nodes[[#]]], {{1, 1, 1}}];deta = Inverse[xy][[All, {1, 2}]]; Kglobal[[#, #]] +=Det[xy]deta.Transpose[deta]/2; F[[#]] += Det[xy]/6) &, elements]; free = Complement[Range[nnode], fixed]; phi[[free]] = LinearSolve[Kglobal[[free, free]], F[[free]]]; Mean[x_] := Plus @@ x/Length[x]; PlotColor[x_] := Hue[2(1 - x)/3]; Show[Graphics[Table[{PlotColor[Mean[phi[[elements[[i]]]]]/Max[phi]], Polygon[nodes[[elements[[i]]]]]}, {i,1, nelem}], AspectRatio -> 1]]; Here is some code for an interpolated plot: (* runtime: 10 seconds *) n = 275; x1 = y1 = -1; x2 = y2 = 1; image = Table[0, {n}, {n}]; xIntersect[{{x1_, y1_}, {x2_, y2_}}] := If[y1 == y2, Infinity, x1 + (y - y1)(x2 - x1)/(y2 - y1)]; Scan[(plist = nodes[[#]]; xlist = nodes[[#, 1]]; ylist = nodes[[#, 2]]; p1 = plist[[1]]; {dx2, dy2} = plist[[2]] - p1; {dx3, dy3} = plist[[3]] - p1; Do[y = y1 + (y2 - y1)(i - 1)/(n - 1); {xa, xb,xc} = Map[xIntersect, Partition[Sort[plist, #1[[2]] < #2[[2]] &], 2, 1, 1]]; jlist = Round[n({xc, If[Abs[xb - xc] < Abs[xa - xc], xb, xa]} - x1)/(x2 - x1)]; If[jlist =!= {}, Do[x = x1 + (x2 - x1)(j - 1)/(n - 1); {xi, eta} = ((x - p1[[1]]){dy3, -dy2} + (y - p1[[2]]){-dx3, dx2})/(dx2 dy3 - dx3 dy2); image[[i, j]] = phi[[#]].{1 - xi - eta, xi, eta}, {j, Max[1, Min[jlist]], Min[n, Max[jlist]]}]], {i, Max[1, Round[n(Min[ylist] - y1)/(y2 - y1)]], Min[n, Round[n(Max[ylist] - y1)/(y2 - y1)]]}]) &, elements]; ListDensityPlot[image, Mesh -> False, Frame -> False, ColorFunction -> PlotColor, Epilog -> Map[{Line[Map[({x, y} = nodes[[#, {1, 2}]]; n{(x - x1)/(x2 -x1), (y - y1)/(y2 - y1)}) &, #]]} &, elements]]; |
Here is some code to solve Poisson’s equation on an unstructured grid of quadrilateral elements:(* runtime: 0.05 second *) n = 10; nodes = Flatten[Table[{x, y}, {x, -1, 1,2.0/(n - 1)}, {y, -1, 1, 2.0/(n - 1)}], 1]; nnode = Length[nodes]; elements =Flatten[Table[{i, i + 1, i + n + 1, i + n} + (j - 1) n, {j, 1, n - 1}, {i, 1, n - 1}], 1]; nelem = Length[elements]; fixed = Map[Position[nodes, #][[1, 1]] &, Select[nodes, Max[Abs[#]] == 1 &]]; Kglobal = Table[0, {nnode}, {nnode}]; F = phi = Table[0, {nnode}]; Scan[(plist = nodes[[#]]; dphi = {plist[[2]] - plist[[1]],plist[[4]] - plist[[1]]}; B = Inverse[Transpose[dphi].dphi]; C1 = {{2, -2}, {-2, 2}}B[[1, 1]] + {{3, 0}, {0, -3}}B[[1, 2]] + {{2, 1}, {1,2}}B[[2, 2]];C2 = {{-1, 1}, {1, -1}}B[[1, 1]] + {{-3, 0}, {0, 3}} B[[1, 2]] + {{-1, -2}, {-2, -1}}B[[2, 2]]; Kglobal[[#, #]] += Det[dphi]Join[Thread[Join[C1, C2]], Thread[Join[C2, C1]]]/6; F[[#]] += Det[Join[{{1, 1, 1}}, Transpose[nodes[[#[[{1, 2, 3}]], {1, 2}]]]]]/4) &, elements]; free = Complement[Range[nnode], fixed]; phi[[free]] = LinearSolve[Kglobal[[free, free]], F[[free]]]; Mean[x_] := Plus @@ x/Length[x]; PlotColor[x_] := Hue[2(1 - x)/3]; Show[Graphics[Table[{PlotColor[Mean[phi[[elements[[i]]]]]/Max[phi]], Polygon[nodes[[elements[[i]]]]]}, {i, 1, nelem}], AspectRatio -> 1]]; Links FEM_50 - simple Matlab Poisson solver for quads and triangles FEM Poisson - paper by Nasser Abbasi Poisson solver - simple finite volume C code by Jeroen Valk |
Finite Element Analysis (FEA) - AutoCAD 2000, AutoLisp, Mathematica 4.2, 5/26/06
Here we see the deformation and stress distribution between two gears where the left gear is fixed but the right gear has an applied torque. This was solved using Finite Element Analysis (FEA) to solve the linear elasticity equations for an unstructured grid of quad elements. The color represents the equivalent (Von Mises) stress.(* runtime: 0.8 second *) n = 18; dtheta = 2Pi/n; nodes = Flatten[Table[r{Sin[theta], Cos[theta]}, {r, 0.4, 1, 0.2}, {theta, 0, 2Pi - dtheta, dtheta}], 1]; elements = Flatten[Table[n j + Append[Table[i + {0,1, n + 1, n}, {i, 1, n - 1}], {1, n + 1, 2n, n}], {j, 0, 2}], 1]; nnode = Length[nodes]; nelem = Length[elements]; i = 3n/2 + 1; fixed = {{i - 1, "x"}, {i, "x"}, {i, "y"}, {i + 1, "x"}}; loads = {{n + 1, {0, -200}}}; Dkl := 0.25{{-1 + y, 1 - y, 1 + y, -1 - y}, {-1 + x, -1 - x, 1 + x, 1 - x}}; Dkm = 0.25{{-1, 1, 1, -1}, {-1, -1, 1, 1}}; e = 1500; nu = 0.3; Epss = {{1, nu, 0}, {nu, 1, 0}, {0, 0, (1 - nu)/2}}e/(1 - nu^2); thk = 1; Kglobal = Table[0, {2nnode}, {2nnode}]; Scan[(element = #; enodes = nodes[[element]]; Klocal = Table[0, {8}, {8}]; Kee = Table[0, {4}, {4}]; Kre = Table[0, {8}, {4}]; Scan[({x, y} = #/Sqrt[3]; t = Inverse[Dkl.enodes].Dkl; strain = {{t[[1, 1]], 0, t[[1, 2]], 0, t[[1, 3]], 0, t[[1, 4]], 0}, {0, t[[2, 1]], 0, t[[2, 2]], 0, t[[2, 3]], 0, t[[2, 4]]}, {t[[2, 1]], t[[1, 1]], t[[2, 2]], t[[1, 2]], t[[2, 3]], t[[1, 3]], t[[2, 4]], t[[1, 4]]}}; ttm = -2Inverse[Dkm.enodes].{{x, 0}, {0, y}}; ct = {{ttm[[1,1]], 0, ttm[[1, 2]], 0}, {0, ttm[[2, 1]], 0, ttm[[2, 2]]}, {ttm[[2, 1]], ttm[[1, 1]], ttm[[2, 2]], ttm[[1, 2]]}}Det[Dkm.enodes]/Det[Dkl.enodes]; Dj = thk Abs[Det[Dkl.enodes]]; Klocal += Transpose[strain].(Epss.strain) Dj; Kee += Transpose[ct].(Epss.ct) Dj; Kre += Transpose[strain].(Epss.ct) Dj) &, {{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}];Klocal -= Kre.Inverse[Kee].Transpose[Kre];Do[Kglobal[[2element[[i]] - di, 2element[[j]] - dj]] += Klocal[[2i -di, 2j - dj]], {di, 0, 1}, {dj, 0, 1}, {i, 1, 4}, {j, 1,4}]) &, elements]; Scan[(i = 2#[[1]] - If[#[[2]] == "x", 1, 0]; Kglobal[[i, i]] *= 10^6) &, fixed]; paload = Table[0, {2nnode}]; Scan[(paload[[2#[[1]] - {1, 0}]] = #[[2]]) &, loads]; del = Partition[LinearSolve[Kglobal, paload], 2]; Do[nodes2 = nodes + scale del; Show[Graphics[Map[Polygon[nodes2[[#]]] &, elements], AspectRatio -> Automatic]], {scale, 0, 1, 0.1}]; |
LinksGears - nice FEA animation by Noran Engineering, Inc. (where I currently work) Mathematica FEA - code for 2D beam using quad elements, by Chee Kiang Yew Mini-FEA - simple Java applet IMTEK Mathematica Supplement - free FEA and meshing packages Rubber Membrane - interesting animation of buckling behavior Physica - has the ability to do “MultiPhysics” (for example, fluid-structure interaction) Real-Time FEA - cool Java applet for 3D beam by Matthias Müller-Fischer AceFEM - FEA environment for Mathematica |
St. Louis Arch - calculated in Mathematica 4.2, rendered in POV-Ray 3.6.1, 10/6/06
The St. Louis Arch is shaped like an upside-down catenary for maximum support. This image was simulated as a system of linear static 1D beam elements. The non-linear animation on the right was simulated using many small linear solutions.(* runtime: 3 seconds *) x0 = 299.2239; h = 625.0925; a = 118.6261; n = 20; EA =1000.0; nodes = Flatten[Table[z = a(Cosh[x0/a] - Cosh[x/a]); r = 54 - 37z/h; Table[{x, 0, z} + r{-Tanh[x/a]Cos[theta], Sin[theta], -Sech[x/a]Cos[theta]}, {theta, 0,4Pi/3, 2Pi/3}], {x, -x0, x0, 2x0/n}], 1]; elements = Flatten[Table[Join[{{1, 2}, {2, 3}, {3, 1}}, If[i < n, {{1, 4}, {2, 5}, {3, 6}, {1, 5}, {2, 6}, {3, 4}}, {}]] + 3i, {i, 0, n}], 1]; nnode = Length[nodes]; nelem = Length[elements]; del = Map[nodes[[#[[2]]]] - nodes[[#[[1]]]] &, elements]; fixed = Join[{1, 2, 3}, nnode - {2, 1, 0}]; nfixed = Length[fixed]; ilist = Flatten[Map[3# - {2, 1, 0} &, Complement[Range[nnode], fixed]], 1]; Kglobal = Sum[d = del[[ielem]]; r = Join[d, -d]; K = EA Map[# r &, r]/(d.d)^1.5; T = Flatten[Table[If[i == 3elements[[ielem, k]] + j, 1, 0], {k, 1, 2}, {j, -2, 0}, {i, 1, 3nnode}], 1]; Transpose[T].K.T, {ielem, 1, nelem}][[ilist, ilist]]; Q = Table[{0, 0, 0}, {nnode - nfixed}]; Q[[Ceiling[nnode/2] - nfixed]] = {0, -1, 0}; q = Partition[LinearSolve[Kglobal, Flatten[Q]], 3]; j = 1; q = Table[If[MemberQ[fixed, i], {0, 0, 0}, q[[j++]]], {i, 1, nnode}]; e = Map[q[[#[[1]]]] - q[[#[[2]]]] &, elements]; s = Map[Sqrt[#.#] &, e]; smax = Max[s]; Do[del2 = del + scale e; nodes2 = nodes; used = Range[nfixed]; Do[Scan[(i2 = elements[[#[[1]], 3 - #[[2]]]]; If[! MemberQ[used, i2],nodes2[[i2]] = nodes2[[i1]] + del2[[#[[1]]]];used = Append[used, i2]]) &, Position[elements, i1]], {i1, 1, nnode}]; Show[Graphics3D[Table[{Hue[2(1 - Min[1, Max[0, scale Abs[s[[i]]]/smax]])/3], Line[nodes2[[elements[[i]]]]]}, {i, 1, nelem}]]], {scale, 0, 1, 0.1}]; Links Deflecting Arch - Mathematica animation by L. Zamiatina Space Truss - Mathematica notebook by Carlos Felippa Sim-POV - mechanics simulations for POV-Ray by Christoph Hormann |
Flapping Wing - Fortran 90, rendered in POV-Ray 3.6.1, 4/28/06
This flapping wing was calculated using the unsteady vortex panel method adapted from Alan Lai’s Fortran code. It assumes inviscid incompressible potential flow (irrotational). I also have a working Mathematica version of this code, but it is a little lengthy to show here.Links Unsteady Panel Code Simulations - by Kevin Jones Insect Flight - Jane Wang made one of the first simulations to predict that an insect can produce sufficient lift to remain aloft. Here is another article about it. See also these Vortex Method Simulations by Jeff Eldredge. Robotic Fly - Uses piezoelectric actuators, by Ron Fearing. See also his Microfly with piezoelectric actuators. Harvard Microrobotics Lab - see this video Flapping MAV - interesting design by Kevin Jones Biomimetics - Biomimetics is the study of biological mechanisms in order to engineer machines that can mimic them. RoboCup - International robotic soccer competitions. Their goal is to build robots that can defeat champion human soccer players by the year 2050. It looks like they still have a way to go. See also RoboGames. Bionic Dolphin - this submarine is too buoyant to go underwater, but when it is going fast enough, its "flippers" work like upside-down airplane "wings" to force it under water. Airic's Arm - bionic arm operated by "Fluidic Muscles" more biomimetics links: Air-Ray (very cool flying manta ray), Aqua-Ray (underwater manta ray), Snake Robot (see video and sidewinding snake), robotic dragonfly toy, NASA's robotic serpent, BigDog, Spinybot II wall climber, RoboPike, robotic caterpillar, RoboSnail, robotic mule, RoboStrider, RoboRoach hexapod, robotic dolphin, RoboLobster |
Joukowski Airfoil - C++, 11/8/07
These animations were created using a conformal mapping technique called the Joukowski Transformation. A Joukowski airfoil can be thought of as a modified Rankine oval. It assumes inviscid incompressible potential flow (irrotational). Potential flow can account for lift on the airfoil but it cannot account for drag because it does not account for the viscous boundary layer (D'Alembert's paradox). In these animations, red represents regions of low pressure. The left animation shows what the surrounding fluid looks like when the Kutta condition is applied. Notice that the fluid separates smoothly at the trailing edge of the airfoil and a low pressure region is produced on the upper surface of the wing, resulting in lift. The lift is proportional to the circulation around the airfoil. The right animation shows what the surrounding fluid looks like when there is no circulation around the airfoil (stall). Notice the sharp singularity at the trailing edge of the airfoil.
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Here is an animation that shows how the streamlines change when you increase the circulation around the airfoil. (Please note: The background fluid motion in this animation is just for effect and is not accurate!) Here is some Mathematica code to plot the streamlines and pressure using Bernoulli’s equation:(* runtime: 13 seconds *) U = rho = 1; chord = 4; thk = 0.5; alpha = Pi/9; y0 = 0.2; x0 = -thk/5.2; L = chord/4; a = Sqrt[y0^2 + L^2]; gamma = 4Pi a U Sin[alpha + ArcCos[L/a]]; w[z_, sign_] := Module[{zeta = (z + sign Sqrt[z^2 - 4 L^2])/2}, zeta = (zeta - x0 - I y0)Exp[-I alpha]/Sqrt[(1 - x0)^2 + y0^2]; U(zeta + a^2/zeta) + I gamma Log[zeta]/(2Pi)]; sign[z_] := Sign[Re[z]]If[Abs[Re[z]] < chord/2 && 0 < Im[z] < 2y0(1 - (2Re[z]/chord)^2), -1, 1];w[z_] := w[z, sign[z]]; V[z_] = D[w[z, sign], z] /. sign -> sign[z]; << Graphics`Master`; DisplayTogether[DensityPlot[-0.5rho Abs[V[(x + I y)Exp[I alpha]]]^2, {x, -3, 3}, {y, -3, 3}, PlotPoints -> 275, Mesh -> False, Frame -> False, ColorFunction -> (If[# == 1, Hue[0, 0, 0], Hue[(5# - 1)/6]] &)],ContourPlot[Im[w[(x + I y)Exp[I alpha]]], {x, -3, 3}, {y, -3, 3}, Contours -> Table[x^3 + 0.0208, {x, -2, 2, 0.1}], PlotPoints -> 100, ContourShading -> False], AspectRatio -> Automatic]; Links Joukowski Animation - nice animation showing how the fluid moves Joukowski Airfoil - nice Java applet by NASA Nikolai Joukowski - used this technique to find the lift on an airfoil in 1906, long before modern computers |
NACA Airfoil - Mathematica 4.2, 12/16/05
Here is a 9415 NACA airfoil. It’s almost shaped the same as the Joukowski airfoil (but it’s a little different):(* runtime: 0.01 second *) n = 40; c = 0.09; p = 0.4; t = 0.15; Clear[x]; camber := c If[x < p, (2p x - x^2)/p^2, ((1 - 2p) + 2p x - x^2)/(1 - p)^2]; theta = ArcTan[D[camber, x]]; p = Table[x = 0.5(1 - Cos[Pi s]); x1 = 1.00893x; thk = 5t(0.2969Sqrt[x1] - 0.126x1 - 0.3516x1^2 + 0.2843x1^3 - 0.1015x1^4); {x, camber} + Sign[s]thk {-Sin[theta],Cos[theta]}, {s, -1, 1, 2/(n - 1)}]; ListPlot[p, PlotJoined -> True, AspectRatio -> Automatic]; We can approximate the pressure profile using the vortex panel method assuming inviscid incompressible potential flow (irrotational). The following Mathematica code was adapted from my Fortran program for my 4/25/99 research project on optimal airfoil design: (* runtime: 0.3 second *) alpha = Pi/9; pc = Table[(p[[i]] + p[[i + 1]])/2, {i, 1, n - 1}]; s = Table[v = p[[i + 1]] - p[[i]];Sqrt[v.v], {i, 1, n - 1}]; theta = Table[v = p[[i + 1]] - p[[i]]; ArcTan[v[[1]], v[[2]]], {i, 1, n - 1}]; sin = Sin[theta]; cos = Cos[theta]; Cn1 = Cn2 = Ct1 = Ct2 =Table[0, {n - 1}, {n - 1}]; Do[If[i == j, Cn1[[i, j]] = -1; Cn2[[i, j]] = 1; Ct1[[i, j]] = Ct2[[i, j]] = Pi/2,v = pc[[i]] - p[[j]]; a = -v.{cos[[j]], sin[[j]]}; b = v.v;t = theta[[i]] - theta[[j]];c = Sin[t]; d = Cos[t]; e = v.{sin[[j]], -cos[[j]]}; f =Log[1 + s[[j]](s[[j]] + 2a)/b]; g = ArcTan[b + a s[[j]], e s[[j]]];t = theta[[i]] - 2theta[[j]]; p1 = v.{Sin[t], Cos[t]}; q = v.{Cos[t], -Sin[t]}; Cn2[[i, j]] = d + (0.5q f - (a c + d e)g)/s[[j]]; Cn1[[i, j]] = 0.5d f + c g - Cn2[[i, j]];Ct2[[i, j]] = c + 0.5p1 f/s[[j]] + (a d - c e)g/s[[j]]; Ct1[[i, j]] = 0.5c f - d g - Ct2[[i, j]]], {i, 1, n - 1}, {j, 1, n - 1}]; gamma = LinearSolve[Table[If[i == n, If[j == 1 || j == n, 1, 0], If[j == n, 0, Cn1[[i, j]]] + If[j == 1, 0, Cn2[[i, j - 1]]]], {i, 1, n}, {j, 1, n}], Table[If[i ==n, 0, Sin[theta[[i]] - alpha]], {i, 1, n}]]; ListPlot[Table[q = Cos[theta[[i]] - alpha] + Sum[(If[j == n, 0, Ct1[[i, j]]] + If[j == 1, 0, Ct2[[i, j - 1]]])gamma[[j]], {j, 1, n}]; {pc[[i, 1]], 1 - q^2}, {i, 1, n - 1}], PlotJoined -> True]; Here is some code for a pretty pressure plot: (* runtime: 33 seconds *) w[z_] := z Exp[-I alpha] + I Sum[s[[j]]gamma[[j]]Log[z - pc[[j, 1]] - I pc[[j, 2]]], {j, 1, n - 1}]; V[z_] = D[w[z], z]; DensityPlot[-Abs[V[(x + I y)Exp[I alpha]]]^2, {x, -0.25, 1.25}, {y, -0.75, 0.75}, PlotPoints -> 275, Mesh -> False, Frame -> False, ColorFunction -> (Hue[(5# - 1)/6] &), AspectRatio -> Automatic]; Links Panel Code - by Kevin Jones Vortex Panel Java applet - by William Devenport |
Four-Stroke Otto Cycle - POV-Ray 3.6.1
under constructionLinks Wankel Engine - a unique engine Engine Assembly - nice FEA animation by Noran Engineering, Inc, based on this model V8 Model - free Solidworks model from 3D ContentCentral |
Differential Transmission - POV-Ray 3.6.1
under constructionLinks Gleason's Differential - a unique design |
Torque Converter - POV-Ray 3.6.1
under construction |
Helicopter Rotor
collective cyclic pitch.Links Palmsize Micro Copter - amazing tiny toy largest helicopters in the world Remote Control Dragonfly - video Tip-Jet Rotors - jet engines on tips of helicopter blades Miniature Osprey - remote control V-22 Osprey Mesicopter - very small helicopter proposed for Mars exporation, by Ilan Kroo and Friedrich Prinz Pixelito - 6.9 gram helicopter Micro Coaxial Rotorcraft - by Felipe Bohorquez, here is a video and an explanation biomimetic rotor - flapping helicopter propeller Chaos maneuver - pitch, roll, yaw Flying Ball Six-Inch MAV - can be launched by throwing it like a frisbee, by Michael Nechyba |
Involute Gears - POV-Ray 3.6.1, 7/12/06
The ideal profile for a gear tooth is the involute of a circle. Here is some Mathematica code:IR = 0.875; OR = 1.125; tmax = Sqrt[OR^2/IR^2 - 1]; theta := Abs[tmax - Mod[t, 2tmax]]; sign := 2Floor[Mod[t/tmax, 2]] - 1; Rotate[{x_, y_}, theta_] := {x Cos[theta] - y Sin[theta], x Sin[theta] + y Cos[theta]}; ParametricPlot[IR Rotate[{Cos[theta] + theta Sin[theta], sign(Sin[theta] - theta Cos[theta])}, Floor[t/(2tmax)]Pi/5 + 0.109152sign], {t, 0, 20.001tmax}, AspectRatio -> 1]; See also my gear orbit trap. Links Gears Trefoil - by Michael Trott PentaGear - gear sphere by Jeffrey Pettyjohn using GearTrax |
Expanding Dodecahedron - POV-Ray 3.6.1, 2/1/07
This dodecahedron unfolds and expands by a factor of 2.61803 (the golden ratio plus one). You could possibly construct something similar to this. I think it would make an interesting display for the entrance to a children’s science museum.Link: Hoberman Sphere - icosidodecahedron toy that expands |
Cube - POV-Ray 3.6.1, 1/31/07
This is a cube that expands by a factor of 3. I think it would be difficult to contruct this.Link: Atomium - amazing 335 foot tall cube building in Brussels built in 1958 |
Torsion Beam - calculated in Mathematica 4.2, rendered in POV-Ray 3.6.1, 5/3/06
Here is an exaggerated example of a beam being twisted. The following code is based on Stephen Timoshenko’s analytical solution. The color represents the equivalent (Von Mises) stress where red is high stress and blue is low stress.(* runtime: 12 seconds *) a = b = 2.0; L = 10; T = G = 1; J = 16/3a^3b(1 - 196a/(b Pi^5)Sum[n^-5Tanh[n Pi b/(2a)], {n, 1, 200, 2}]); k = T/(G J); phi = 32G k a/Pi^3Sum[1/n^3(-1)^((n - 1)/2)(1 - Cosh[n Pi y/(2a)]/Cosh[n Pi b/(2a)])Cos[n Pi x/(2a)], {n, 1, 10, 2}]; tauyz = -D[phi, x]; tauxz = D[phi, y]; tauxy = sx = sy = sz = 0; f[x_, y_, z_] := Module[{}, {s1, s2, s3} = Eigenvalues[{{sx, tauxy, tauxz}, {tauxy, sy, tauyz}, {tauxz, tauyz, sz}}]; sv = Sqrt[((s1 - s2)^2 + (s2 - s3)^2 + (s3 - s1)^2)/2]; {x - k z y, y + k z x, z, {EdgeForm[], SurfaceColor[Hue[2(1 - sv/0.029)/3]]}}]; << Graphics`Master`; DisplayTogether[Map[{ParametricPlot3D[f[#a/2, y, z], {y, -b/2, b/2}, {z, 0,L}, Compiled -> False], ParametricPlot3D[f[x, #b/2, z], {x, -a/2, a/2}, {z, 0, L}, Compiled -> False]} &, {-1, 1}], ParametricPlot3D[f[x, y, L], {x, -a/2, a/2}, {y, -b/2, b/2}, Compiled -> False]]; Links Computer Modeling - class notes by Nasser Abbasi Torsional Analysis - part of the Structural Mechanics Mathematica package |
Solution to Poisson’s equation on Structured Grid - calculated in GridPro, rendered in POV-Ray 3.6.1, 10/5/06
Here is a structured hexahedral mesh. We use GridPro for generating complicated grids for our CFD applications. GridPro optimizes the grid spacing by placing gridpoints along isocontours of the solution to the Poisson equation. |
Unstructured Tetrahedral Meshed Sphere - DistMesh, Matlab, POV-Ray 3.6.1, 6/12/06
Here is an unstructured tetrahedral meshed sphere.Links Mesh Generation - interesting paper by Per-Olof Persson Free Mesh Generators - click here for a complete list: CUBIT (not free) - impressive hexahedral mesh generator |
Folding Wing Aircraft
calculated in CFL3D (Computational Fluids Laboratory 3D), rendered in POV-Ray 3.6.1, 11/3/06
This is a preliminary test of a proposed project to simulate the wing fluttering of an aircraft with folding wings. This was solved using the finite volume method for 3D, steady, compressible, viscous, laminar flow on a curvilinear grid. This image shows the pressure distribution over the wings.Links papers - wing flutter and gridless methods by Feng Liu |
Supersonic Flow - CFL3D (Computational Fluids Laboratory 3D), Tecplot 360, 2/1/07
Here are some weird tests of CFL3D. The color represents the Mach number.Links Shock Diamonds |
Simply Loaded Beam - Patran, Nastran (NASA Structural Analysis), 9/28/06
Nastran was originally created by NASA but now other companies are distributing it. This was a simple class assignment. I’m planning to post a more interesting picture here later. |
Other Engineering Links
The Future of Things (TFOT) - tons of high-tech gadgets
Freedom Ship - proposed ship, 3 million tons, 4500 feet long
Habbakuk - proposed Pykrete aircraft carrier during World War II, 2 million tons, 2000 feet long
Space Elevator - made from carbon nanotubes
Microsubmarine
PowerLabs (Sam Barros) - impressive projects, see the rail gun, jet turbine, and high speed CD-Rom experiments
Motorized Surfboard
Flexinol - wires that electronically contract like muscles, sample kit was $30 last time I checked
Air Muscle - interesting actuator
Holographic Flat Panel - uses parallax to create a 3D effect, no glasses required
World’s Smallest Internal Combustion Engine
wind-powered art sculpture - weighs nearly 3 tons with living quarters inside
Cloud-busting - how to make it rain using liquid nitrogen
Architecture Links
Extreme Engineering - interesting Discovery web site on ambitious architectural plans
Skyscraper Page - over 16000 scale drawings of famous buildings
The Palm Island - man-made island in Dubai
ETFE (Ethylene Tetrafluoroethylene) - new high strength insulating plastic, check out this amazing picture