CS-184: Computer Graphics. Today. Lecture #22: Spring and Mass systems. Spring and Mass systems. Distance springs Spring dampers Edge springs
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1 CS-184: Computer Graphics Lecture #: Spring and Mass systems Prof. James O Brien University of California, Berkeley V011-F--1.0 Today Spring and Mass systems Distance springs Spring dampers Edge springs
2 Huamin Wang, Ravi Ramamoorthi, and James F. O'Brien. "Data-Driven Elastic Models for Cloth: Modeling and Measurement". ACM Transactions on Graphics, 30(4):71:1 11, July 011. Proceedings of ACM SIGGRAPH 011, Vancouver, BC Canada. 3 Huamin Wang, Ravi Ramamoorthi, and James F. O'Brien. "Data-Driven Elastic Models for Cloth: Modeling and Measurement". ACM Transactions on Graphics, 30(4):71:1 11, July 011. Proceedings of ACM SIGGRAPH 011, Vancouver, BC Canada. 4
3 A Simple Spring Ideal zero-length spring f a!b = ks(b a) Force pulls points together f b!a = Strength proportional to distance f a!b 5 A Simple Spring Energy potential E =1/ ks(b a) (b a) f a!b = ks(b a) f b!a = f a!b fa = rae 6
4 A Simple Spring Energy potential: kinetic vs elastic E =1/ ks(b a) (b a) E =1/ m(ḃ ȧ) (ḃ ȧ) -1 7 Non-Zero Length Springs 3 f a!b = ks b a ( b a l) b a Rest length E = ks ( b a l)
5 Comments on Springs Springs with zero rest length are linear Springs with non-zero rest length are nonliner Force magnitude linear w/ discplacement (from rest length) Force direction is non-linear Singularity at b a =0 9 Damping Mass proportional damping f ȧ f = k d ȧ Behaves like viscous drag on all motion Consider a pair of masses connected by a spring How to model rusty vs oiled spring Should internal damping slow group motion of the pair? Can help stability... up to a point 10
6 Damping Stiffness proportional damping b a fa = k d b a (b a) (ḃ ȧ) Behaves viscous drag on change in spring length Consider a pair of masses connected by a spring How to model rusty vs oiled spring Should internal damping slow group motion of the pair? 11 Spring Constants Two ways to model a single spring l l/ l/ l/ l l/ 1
7 Spring Constants ks Constant gives inconsistent results with different discretizations Change in length is not what we want to measure Strain: change in length as fraction of original length = l 0 l Nice and simple for 1D Structures from Springs Sheets Blocks Others 14
8 Structures from Springs They behave like what they are (obviously!) This structure will not resist shearing This structure will not resist outof-plane bending either Structures from Springs They behave like what they are (obviously!) This structure will resist shearing but has anisotopic bias This structure still will not resist out-of-plane bending 16
9 Structures from Springs They behave like what they are (obviously!) This structure will resist shearing Less bias Interference between spring sets This structure still will not resist out-of-plane bending 17 Structures from Springs They behave like what they are (obviously!) This structure will resist shearing Less bias Interference between spring sets This structure will resist out-ofplane bending Interference between spring sets Odd behavior How do we set spring constants? 18
10 Edge Springs 3 ^ e 4 ^ n1 ^ n Figure 1: 1 ^n 1 ^ n π N u 1 = E 1 N N 1 u = E N u 3 = (x 1 x 4 ) E E u 4 = (x 1 x 3 ) E E N 1 N 1 + (x x 4 ) E E N 1 N 1 (x x 3 ) E E N N N N to an arbitrary scaling factor, where N = (x x ) F e i = k e E N 1 + N sin(θ/)u i, 4. The elastic bending stiffness From Bridson et al., 003, also see Grinspun et al., Example: Thin Material Discrete Shells SCA 003 Eitan Grinspun, Anil Hirani, Mathieu Desbrun and Peter Schröder 0
11 Strain Limiting Huamin Wang, James F. O'Brien, and Ravi Ramamoorthi. "Multi-Resolution Isotropic Strain Limiting". In Proceedings of ACM SIGGRAPH Asia 010, pages 160:1 10, December Suggested Reading Physically Based Modeling: Principles and Practice Andy Witkin and David Baraff Grinspun, Hirani, Desbrun, and Peter Schroder, "Discrete Shells," SCA 003 Bridson, Marino, and Fedkiw, "Simulation of Clothing with Folds and Wrinkles," SCA 003 O Brien and Hodgins, Graphical Modeling and Animation of Brittle Fracture, SIGGRAPH 99
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