A modified HHT method for the numerical simulation of rigid body rotations with Euler parameters

Abstract

In multibody dynamics, the Euler parameters are often used for the numerical simulation of rigid body rotations because they lead to a relatively simple form of the rotation matrix which avoids the evaluation of trigonometric functions and can thus save computational time. The Newmark method and the closely related Hilber-Hughes-Taylor (HHT) method are widely employed for solving the equations of motion of mechanical systems. They can also be applied to constrained systems described by differential algebraic equations. However, in the classical versions, the use of these integration schemes have a very unfavorable impact on the Euler parameter description of rotational motions. In this paper, we show analytically that the angular velocity for a rotation about a single axis under a constant moment will not increase linearly but grows slower. This effect, which does not appear for Euler angles, can be even observed if the numerical damping parameter α in the HHT method is set to zero. To circumvent this problem without losing the advantage of Euler parameters, we present a modified HHT method which reduces the damping effect on the angular velocity significantly and eliminates it completely for α = 0.

Simulation of high speed rotating dynamics in constrained mechanical systems

“…Meanwhile, it can be derived that the angular velocity and the local rotation vector satisfy where After obtaining the value of the local rotation vector, the rotation matrix of the next step A n + 1 can be computed by For a comparison, the DAE system will also be solved through a set of corresponding Euler-Poincaré equations as follows. The dynamic equations are the first two equations in equation (1); the kinematic equations are where bold-italice = false( e 0 , .1em e 1 , .1em e 2 , .1em e 3 false) describes the Euler parameters; the transformation matrix boldE is and rotation matrix boldA is written as 22,1 where and the Lagrange multiplier ν corresponds to the normalized constraint of the Euler parameters and μ is the kinematic Lagrange multiplier corresponding to the constraint equations in equation (2); ν corresponds to the normalized constraint of the Euler parameters; and all the constraint equations can be expressed as The first two equations in Eq. (1) together with Eqs.…”

“…For example, when using Lagrange equation with Euler parameters, the angular velocity of a rigid body was observed not to increase linearly but grows slower until reaching a saturation value under a constant torque. 1,2 This phenomenon, however, can be alleviated by a modification on the HHT integrator that guarantees the angular velocity vector to stay in the Lie algebra s o false( 3 false), 1 or by using the Euler-Poincaré equations. 2 Another approach is to solve the dynamics by Lie group methods.…”

Application of Gauss principle of least constraint in multibody systems with redundant constraints

“…Using equation (9), the dynamic equations of motion turn into a set of ordinary differential equations (ODEs), which could be solved by some numerical integration methods, e.g. the backward differentiation formulas 20 , and the Hilber–Hughes–Taylor method 21 . However, the absence of constraint equations of velocity and position will lead to the constraint violation caused by accumulated numerical integration errors.…”

“…If the Lagrange multipliers vector k is known in advance, equation (21) will always have the unique solution, because the leading matrix of equation 21is always positive definite regardless of the redundant constraints. An iterative procedure for each integration time step is obtained by merely comparing equation 21with equation 9, as follows…”

New Insights into a Three-Sub-Step Composite Method and Its Performance on Multibody Systems

“…The asymptotic annihilating (or L-stable) BDF is particularly useful for stiff problems, whereas it cannot automatically start [11]. The controllably dissipative (or A-stable) generalized-α method and HHT-α method are more popular in multibody systems [16][17][18][19][20]. However, they both use the time weighted residual representation of motion equations, resulting in their acceleration being first-order accurate [21].…”