rmation?" I know that this can already be accomplished using the brilliant Kangaroo plugin, but I wanted a simpler and faster (yet still accurate) single component that could replicate this unique curve using a variety of inputs: the length of the rod/wire, the width/distance between the endpoints, the height of the bend, and the tangent angle at the start. I also wanted make the unknowns (such as height if only length and width are known) easily accessible for plugging into additional components.
The resulting script, being an all-in-one solution, is somewhat unwieldy, but it could easily be broken down into smaller components (custom .gha's which I don't have the ability to code). If someone wants to tackle this, please do! I'm not an expert coder by any means, and as this was only my second time diving into Grasshopper scripting, if the script seems somewhat strange, that's probably why. I did try to comment the code pretty well though. Here's the full description:
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DESCRIPTION: This beast creates the so-called 'elastica curve', the shape a long, thin rod or wire makes when it is bent elastically (i.e. not permanently). In this case, force is assumed to only be applied horizontally (which would be in line with the rod at rest) and both ends are assumed to be pinned or hinged meaning they are free to rotate (as opposed to clamped, when the end tangent angle is fixed, usually horizontally). An interesting finding is that it doesn't matter what the material or cross-sectional area is, as long as they're uniform along the entire length. Everything makes the same shape when bent as long as it doesn't cross the threshold from elastic to plastic (permanent) deformation (I don't bother to find that limit here, but can be found if the yield stress for a material is known).
Key to the formulas used in this script are elliptic integrals, specifically K(m), the complete elliptic integral of the first kind, and E(m), the complete elliptic integral of the second kind. There was a lot of confusion over the 'm' and 'k' parameters for these functions, as some people use them interchangeably, but they are not the same. m = k^2 (thus k = Sqrt(m)). I try to use the 'm' parameter exclusively to avoid this confusion. Note that there is a unique 'm' parameter for every configuration/shape of the elastica curve.
This script tries to find that unique 'm' parameter based on the inputs. The algorithm starts with a test version of m, evaluates an expression, say 2*E(m)/K(m)-1, then compares the result to what it should be (in this case, a known width/length ratio). Iterate until the correct m is found. Once we have m, we can then calculate all of the other unknowns, then find points that lie on that curve, then interpolate those points for the actual curve. You can also use Wolfram|Alpha as I did to find the m parameter based on the equations in this script (example here: http://tiny.cc/t4tpbx for when say width=45.2 and length=67.1).
Other notes:
* This script works with negative values for width, which will creat a self-intersecting curve (as it should). The curvature of the elastica starts to break down around m=0.95 (~154°), but this script will continue to work until M_MAX, m=0.993 (~169°). If you wish to ignore self-intersecting curves, set ignoreSelfIntersecting to True
* When the only known values are length and height, it is actually possible for certain ratios of height to length to have two valid m values (thus 2 possible widths and angles). This script will return them both.
* Only the first two valid parameters (of the required ones) will be used, meaning if all four are connected (length, width or a PtB, height, and angle), this script will only use length and width (or a PtB).
* Depending on the magnitude of your inputs (say if they're really small, like if length < 10), you might have to increase the constant ROUNDTO at the bottom
REFERENCES: {1} "The elastic rod" by M.E. Pacheco Q. & E. Pina, http://www.scielo.org.mx/pdf/rmfe/v53n2/v53n2a8.pdf {2} "An experiment in nonlinear beam theory" by A. Valiente, http://www.deepdyve.com/lp/doc/I3lwnxdfGz {3} "Snap buckling, writhing and Loop formation In twisted rods" by V.G.A. GOSS, http://myweb.lsbu.ac.uk/~gossga/thesisFinal.pdf {4} "Theory of Elastic Stability" by Stephen Timoshenko, http://www.scribd.com/doc/50402462/Timoshenko-Theory-of-Elastic-Stability (start on p. 76)
INPUT: PtA - First anchor point (required) PtB - Second anchor point (optional, though 2 out of the 4--length, width, height, angle--need to be specified) [note that PtB can be the same as PtA (meaning width would be zero)] [also note that if a different width is additionally specified that's not equal to the distance between PtA and PtB, then the end point will not equal PtB anymore] Pln - Plane of the bent rod/wire, which bends up in the +y direction. The line between PtA and PtB (if specified) must be parallel to the x-axis of this plane
** 2 of the following 4 need to be specified ** Len - Length of the rod/wire, which needs to be > 0 Wid - Width between the endpoints of the curve [note: if PtB is specified in addition, and distance between PtA and PtB <> width, the end point will be relocated Ht - Height of the bent rod/wire (when negative, curve will bend downward, relative to the input plane, instead) Ang - Inner departure angle or tangent angle (in radians) at the ends of the bent rod/wire. Set up so as width approaches length (thus height approaches zero), angle approaches zero
* Following variables only needed for optional calculating of bending force, not for shape of curve. E - Young's modulus (modulus of elasticity) in GPa (=N/m^2) (material-specific. for example, 7075 aluminum is roughly 71.7 GPa) I - Second moment of area (or area moment of inertia) in m^4 (cross-section-specific. for example, a hollow rod would have I = pi * (outer_diameter^4 - inner_diameter^4) / 32 Note: E*I is also known as flexural rigidity or bending stiffness
OUTPUT: out - only for debugging messages Pts - the list of points that approximate the shape of the elastica Crv - the 3rd-degree curve interpolated from those points (with accurate start & end tangents) L - the length of the rod/wire W - the distance (width) between the endpoints of the rod/wire H - the height of the bent rod/wire A - the tangent angle at the (start) end of the rod/wire F - the force needed to hold the rod/wire in a specific shape (based on the material properties & cross-section) **be sure your units for 'I' match your units for the rest of your inputs (length, width, etc.). Also note that the critical buckling load (force) that makes the rod/wire start to bend can be found at height=0
THANKS TO: Mårten Nettelbladt (thegeometryofbending.blogspot.com) Daniel Piker (Kangaroo plugin) David Rutten (Grasshopper guru) Euler & Bernoulli (the O.G.'s)
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Edit: More on the math behind this here.
Cheers,
Will
…
Added by Will McElwain at 4:08pm on February 26, 2014
The PC actually stops working because after a few seconds the simulation starts the fan inside the PC all of a sudden stops and for the next 5-10 mins I cannot do anything, even alt+ctrl+canc. After I wait for that time i get the followig error:
the ReadMe says:
{0;0;0}0. Grid-based Radiance simulation1. The component is checking ad, as, ar and aa values. This is just to make sure that the results are accurate enough.2. -ar is set to 300.3. Good to go!4. Current working directory is set to: C:\Users\Luigi\Desktop\Prova__\Prova_1\gridBasedSimulation\5. Found a trans material... Resetting st parameter from 0.85 to 0.011276004966. WMIC PROCESS get Commandline7. WMIC PROCESS get Commandline8. WMIC PROCESS get Commandline9. WMIC PROCESS get Commandline10. WMIC PROCESS get Commandline11. WMIC PROCESS get Commandline12. WMIC PROCESS get Commandline13. WMIC PROCESS get Commandline14. WMIC PROCESS get Commandline15. WMIC PROCESS get Commandline16. WMIC PROCESS get Commandline17. WMIC PROCESS get Commandline18. WMIC PROCESS get Commandline19. WMIC PROCESS get Commandline20. WMIC PROCESS get Commandline21. WMIC PROCESS get Commandline22. WMIC PROCESS get Commandline23. WMIC PROCESS get Commandline24. WMIC PROCESS get Commandline25. WMIC PROCESS get Commandline26. WMIC PROCESS get Commandline27. WMIC PROCESS get Commandline28. WMIC PROCESS get Commandline29. WMIC PROCESS get Commandline30. WMIC PROCESS get Commandline31. WMIC PROCESS get Commandline32. WMIC PROCESS get Commandline33. WMIC PROCESS get Commandline34. WMIC PROCESS get Commandline35. WMIC PROCESS get Commandline36. WMIC PROCESS get Commandline37. WMIC PROCESS get Commandline38. WMIC PROCESS get Commandline39. WMIC PROCESS get Commandline40. WMIC PROCESS get Commandline41. WMIC PROCESS get Commandline42. WMIC PROCESS get Commandline43. WMIC PROCESS get Commandline44. WMIC PROCESS get Commandline45. WMIC PROCESS get Commandline46. WMIC PROCESS get Commandline47. WMIC PROCESS get Commandline48. WMIC PROCESS get Commandline49. WMIC PROCESS get Commandline50. WMIC PROCESS get Commandline51. WMIC PROCESS get Commandline52. WMIC PROCESS get Commandline53. WMIC PROCESS get Commandline54. WMIC PROCESS get Commandline55. WMIC PROCESS get Commandline56. WMIC PROCESS get Commandline57. WMIC PROCESS get Commandline58. WMIC PROCESS get Commandline59. WMIC PROCESS get Commandline60. WMIC PROCESS get Commandline61. WMIC PROCESS get Commandline62. WMIC PROCESS get Commandline63. WMIC PROCESS get Commandline64. WMIC PROCESS get Commandline65. WMIC PROCESS get Commandline66. WMIC PROCESS get Commandline67. WMIC PROCESS get Commandline68. WMIC PROCESS get Commandline69. WMIC PROCESS get Commandline70. WMIC PROCESS get Commandline71. WMIC PROCESS get Commandline72. WMIC PROCESS get Commandline73. WMIC PROCESS get Commandline74. WMIC PROCESS get Commandline75. WMIC PROCESS get Commandline76. WMIC PROCESS get Commandline77. WMIC PROCESS get Commandline78. WMIC PROCESS get Commandline79. WMIC PROCESS get Commandline80. WMIC PROCESS get Commandline81. WMIC PROCESS get Commandline82. WMIC PROCESS get Commandline83. WMIC PROCESS get Commandline84. WMIC PROCESS get Commandline85. WMIC PROCESS get Commandline86. WMIC PROCESS get Commandline87. WMIC PROCESS get Commandline88. WMIC PROCESS get Commandline89. WMIC PROCESS get Commandline90. WMIC PROCESS get Commandline91. WMIC PROCESS get Commandline92. WMIC PROCESS get Commandline93. WMIC PROCESS get Commandline94. WMIC PROCESS get Commandline95. WMIC PROCESS get Commandline96. WMIC PROCESS get Commandline97. WMIC PROCESS get Commandline98. WMIC PROCESS get Commandline99. WMIC PROCESS get Commandline100. WMIC PROCESS get Commandline101. WMIC PROCESS get Commandline102. WMIC PROCESS get Commandline103. WMIC PROCESS get Commandline104. WMIC PROCESS get Commandline105. WMIC PROCESS get Commandline106. WMIC PROCESS get Commandline107. WMIC PROCESS get Commandline108. WMIC PROCESS get Commandline109. WMIC PROCESS get Commandline110. WMIC PROCESS get Commandline111. WMIC PROCESS get Commandline112. WMIC PROCESS get Commandline113. WMIC PROCESS get Commandline114. WMIC PROCESS get Commandline115. WMIC PROCESS get Commandline116. WMIC PROCESS get Commandline117. WMIC PROCESS get Commandline118. WMIC PROCESS get Commandline119. WMIC PROCESS get Commandline120. WMIC PROCESS get Commandline121. WMIC PROCESS get Commandline122. WMIC PROCESS get Commandline123. WMIC PROCESS get Commandline124. WMIC PROCESS get Commandline125. WMIC PROCESS get Commandline126. WMIC PROCESS get Commandline127. WMIC PROCESS get Commandline128. WMIC PROCESS get Commandline129. WMIC PROCESS get Commandline130. WMIC PROCESS get Commandline131. WMIC PROCESS get Commandline132. WMIC PROCESS get Commandline133. WMIC PROCESS get Commandline134. WMIC PROCESS get Commandline135. WMIC PROCESS get Commandline136. WMIC PROCESS get Commandline137. WMIC PROCESS get Commandline138. WMIC PROCESS get Commandline139. WMIC PROCESS get Commandline140. WMIC PROCESS get Commandline141. WMIC PROCESS get Commandline142. WMIC PROCESS get Commandline143. WMIC PROCESS get Commandline144. WMIC PROCESS get Commandline145. WMIC PROCESS get Commandline146. WMIC PROCESS get Commandline147. WMIC PROCESS get Commandline148. WMIC PROCESS get Commandline149. WMIC PROCESS get Commandline150. WMIC PROCESS get Commandline151. WMIC PROCESS get Commandline152. WMIC PROCESS get Commandline153. WMIC PROCESS get Commandline154. WMIC PROCESS get Commandline155. WMIC PROCESS get Commandline156. WMIC PROCESS get Commandline157. WMIC PROCESS get Commandline158. WMIC PROCESS get Commandline159. WMIC PROCESS get Commandline160. WMIC PROCESS get Commandline161. WMIC PROCESS get Commandline162. WMIC PROCESS get Commandline163. WMIC PROCESS get Commandline164. WMIC PROCESS get Commandline165. WMIC PROCESS get Commandline166. WMIC PROCESS get Commandline167. WMIC PROCESS get Commandline168. WMIC PROCESS get Commandline169. WMIC PROCESS get Commandline170. WMIC PROCESS get Commandline171. WMIC PROCESS get Commandline172. WMIC PROCESS get Commandline173. WMIC PROCESS get Commandline174. WMIC PROCESS get Commandline175. WMIC PROCESS get Commandline176. WMIC PROCESS get Commandline177. WMIC PROCESS get Commandline178. WMIC PROCESS get Commandline179. WMIC PROCESS get Commandline180. WMIC PROCESS get Commandline181. WMIC PROCESS get Commandline182. WMIC PROCESS get Commandline183. WMIC PROCESS get Commandline184. WMIC PROCESS get Commandline185. WMIC PROCESS get Commandline186. WMIC PROCESS get Commandline187. WMIC PROCESS get Commandline188. WMIC PROCESS get Commandline189. WMIC PROCESS get Commandline190. WMIC PROCESS get Commandline191. WMIC PROCESS get Commandline192. WMIC PROCESS get Commandline193. WMIC PROCESS get Commandline194. WMIC PROCESS get Commandline195. WMIC PROCESS get Commandline196. WMIC PROCESS get Commandline197. WMIC PROCESS get Commandline198. Runtime error (IndexOutOfRangeException): index out of range: 0199. Traceback: line 320, in script
The thing is that if I raise the -aa parameter from 0.05 to 0.1 all works fine..
Is this only related to my PC then?? What should I do to solve this issue?
Thanks again for your help
Luigi…
思った感じになりません。
balls の代わりにplanarカーブを直接入れてみましたがエラーが出ます。
ファンクションにしてみたところ、forループので作った数値が反映されていません。
ファンクションのインスタンス?を出力していないと思い上記のようにしましたがエラーが出てしまいます。
以上の事から自分の認識が正しいのかよくわからなくなりました・・・
python自体の深いところをわかっているわけではないので余計こんがらがりました。
そこで、for b in ballsはどのような条件または使い方であれば使えるのでしょうか?
そして、上記のように別のオブジェクトに対しての使い方はどのようにすればできるのでしょうか?
2:同じファンクション内のdist = rs.Distance(self.pos,b.pos)についてですが
この文章も for b in balls によってbはBallのインスタンスであると定義?されたためb.posがbの位置であると分かるのでしょうか?
pythonは定義しなくても動いてしまうのでどのような時に使えるのか文章見ただけではよくわかりません・・・
大変細かいことかもしれませんが、よりpythonをしっかりと理解するためにも、どなたかわかる方ご教授いただけると幸いです。…
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Students need to bring their own laptops, digital equipment and model making tools.…