A Lagrangian method for constrained dynamics in tensegrity systems with compressible bars
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ORIGINAL PAPER
A Lagrangian method for constrained dynamics in tensegrity systems with compressible bars Shao-Chen Hsu1 · Vaishnav Tadiparthi2
· Raktim Bhattacharya2
Received: 16 March 2020 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents a Lagrangian approach to simulating multibody dynamics in a tensegrity framework with an ability to tackle holonomic constraint violations in an energy-preserving scheme. Governing equations are described using nonminimum coordinates to simplify descriptions of the structure’s kinematics. To minimize constraint drift arising from this redundant system, the direct correction method has been employed in conjunction with a novel energy-correcting scheme that treats the total mechanical energy of the system as a supplementary constraint. The formulation has been extended to allow tensegrity structures with compressible bars, allowing for further discussion on potential choices for softer bar materials. The benchmark example involving a common tensegrity structure demonstrates the superiority of the presented formulation over Simscape Multibody in terms of motion accuracy as well as energy conservation. The effectiveness of the energy correction scheme is found to be increasing with the extent of deformations in the structure. Keywords Multibody dynamics · Tensegrity · Non-minimum coordinates · Direct correction method · Energy-preserving scheme · Compressible bars
List of symbols Poisson’s ratio of kth bar material (comνk pressible) Force density of kth bar (compressible) Ψk Force density of kth string σk λ Lagrange multipliers Angular momentum of kth bar hk B Bar matrix kth bar bk Connectivity matrix of bars Cb Connectivity matrix of strings Cs F Non-conservative force matrix Damper force in kth string f d,k
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Vaishnav Tadiparthi [email protected] Shao-Chen Hsu [email protected] Raktim Bhattacharya [email protected]
1
Genesys Aerosystems, Mineral Wells, TX 76067, USA
2
Intelligent Systems Research Laboratory, Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA
ωk I bk L pm N ni P pk q R(q) S sk c E K bk K sk l bk l sk rk T Vg Vs Wf
Angular velocity of kth bar Moment of inertia of kth bar Location matrix of point masses Nodal matrix describing the tensegrity structure Position of ith node Point mass matrix kth point mass Coordinates in vector form Ideal constraints String matrix kth string Damping coefficient Total energy of the system Stiffness of kth bar (compressible) Stiffness of kth string Length of kth bar Natural length of kth string Radius of kth bar (compressible) Total kinetic energy Potential energy due to gravity Potential energy of strings Work done by force f
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Computational Mechanics
1 Introduction A tensegrity system is an arrangement of axially-loaded elements (no element bends, even though the overall structure bends), that we loosely characterize as a network of bars and cables. The bars take compressive axial loads and the cables handle tensile loads.
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