# American Institute of Mathematical Sciences

• Previous Article
Study of adaptive symplectic methods for simulating charged particle dynamics
• JCD Home
• This Issue
• Next Article
Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor
December  2019, 6(2): 409-427. doi: 10.3934/jcd.2019021

## Chains of rigid bodies and their numerical simulation by local frame methods

 1 Nordre Krabbedalen 17, 5178 Loddefjord, Norway 2 Matematisk Institutt, Universitetet i Bergen, 5020 Bergen, Norway

* Corresponding author: Antonella Zanna

Received  March 2019 Revised  September 2019 Published  November 2019

We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [18]. In this framework, the dynamics of the $j$th body is described in a frame relative to the $(j-1)$th one. Starting from the Lagrangian formulation of the system on ${{\rm{SO}}}(3)^{N}$, the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.

Citation: Nicolai Sætran, Antonella Zanna. Chains of rigid bodies and their numerical simulation by local frame methods. Journal of Computational Dynamics, 2019, 6 (2) : 409-427. doi: 10.3934/jcd.2019021
##### References:

show all references

##### References:
System of chained rigid bodies
Single solid pendulum, consisting of a rectangular prism. Left: classical planar motion, described by the angle $\theta$. Right: planar motion, described in 3D by a quaternion and the MFM. At rest (vertical position), the inertial and frame axes are aligned
Numerical solution for the angle $\theta$ for the solid pendulum. The MFM is solved with IMR and stepsize $h = 0.01$ and the exact solution is computed with RK45 imposing machine precision on the tolerances
A simulation of the spinning top, idealized as a rod with a disk (left). Right: position of the center of mass in $[0,30]$ integrated with step size $h = 0.01$. See text for the values of the remaining parameters. The formulation of the equations is identical to that of the single pendulum
4-body pendulum with torsional springs
16-body
Conservation of quaternion-unit length (left) and relative energy error (right) for the 16-body pendulum solved with GLRK scheme with a step length $h = 0.01$
Spring loaded/damped chain. The red vector represents $n$, the unit vector in the direction of the displacement of the spring
64-body pendulum
Comparison of the error in unit length (for quaternions), relative energy error and function evaulations at $T = 10$ for the RK45 and 2-stages GLRK method for a 3D pendulum with 4 links. See text for more details
 Link 1 Link 2 Link 3 Link 4 Relative energy error Function evaluations RK45 5.55e-15 2.47e-14 2.24e-14 2.50e-14 7.22e-14 15794 GL2 2.22e-15 5.55e-16 3.00e-15 1.66e-15 3.00e-13 11334
 Link 1 Link 2 Link 3 Link 4 Relative energy error Function evaluations RK45 5.55e-15 2.47e-14 2.24e-14 2.50e-14 7.22e-14 15794 GL2 2.22e-15 5.55e-16 3.00e-15 1.66e-15 3.00e-13 11334
Computational times for systems with $N$ links. The computational cost grows quadratically with the number of links. See text for more details
 $N$ 4 8 16 32 48 64 Time 719ms$\pm$4.31ms 2.27s$\pm$15.8ms 10.7s$\pm$122ms 45.6$\pm$1.55s 1m43s$\pm$1.94s 3m24s$\pm$5.52s
 $N$ 4 8 16 32 48 64 Time 719ms$\pm$4.31ms 2.27s$\pm$15.8ms 10.7s$\pm$122ms 45.6$\pm$1.55s 1m43s$\pm$1.94s 3m24s$\pm$5.52s
 [1] Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021020 [2] Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002 [3] Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences. Electronic Research Archive, , () : -. doi: 10.3934/era.2021025 [4] F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605 [5] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [6] Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021048 [7] Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 [8] Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022 [9] Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $\Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 [10] Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 [11] Youjun Deng, Hongyu Liu, Xianchao Wang, Dong Wei, Liyan Zhu. Simultaneous recovery of surface heat flux and thickness of a solid structure by ultrasonic measurements. Electronic Research Archive, , () : -. doi: 10.3934/era.2021027 [12] Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042 [13] Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407 [14] Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007 [15] Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 [16] Yongming Luo, Athanasios Stylianou. On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3455-3477. doi: 10.3934/dcdsb.2020239 [17] Hao Li, Honglin Chen, Matt Haberland, Andrea L. Bertozzi, P. Jeffrey Brantingham. PDEs on graphs for semi-supervised learning applied to first-person activity recognition in body-worn video. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021039 [18] Miroslav Bulíček, Victoria Patel, Endre Süli, Yasemin Şengül. Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021053 [19] Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $m$-ary $n$-cubes $Q^m_n$. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079 [20] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

Impact Factor: