Abstract
The Euler methods on Lie group are developed for the differential–algebraic equations of multibody system dynamics with holonomic constraints. The implicit Euler method is used to solve the differential–algebraic equations as Euler–Lagrange equations on Lie group with indices 1, 2, and 3 and the case of overdetermined differential–algebraic equations mixing with configuration space. The symplectic Euler method is used to solve the differential–algebraic equations as constrained Hamilton equations on Lie group. For the discrete mapping between Lie group and Lie algebra, the canonical coordinates of the second kind for implicit first-order Crouch–Grossman Euler methods of differential–algebraic equations are used. A single pendulum and a double pendulum in the space are used to verify the accuracy of the Lie group Euler methods.
Highlights
Geometric integration methods[1,2,3,4] have received considerable attention in computational multibody system dynamics due to their numerical stabilities
Results illustrate that the methods DAE3-IEL and overdetermined differential–algebraic equations (ODAEs)-IEL have the similar accuracy of obtaining the total energy and constraints
The methods DAE2-IELS and DAE2-SELS have the similar accuracy of obtaining the total energy, while DAE2IELS has the higher accuracy of obtaining the constraints
Summary
Geometric integration methods[1,2,3,4] have received considerable attention in computational multibody system dynamics due to their numerical stabilities. One can get the implicit Euler method of DAEs (equation (1)) with index 3 on Lie group as Using equations (13) and (10), the implicit Euler method of DAEs (equation (4)) with index 2 on Lie group is
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