1. Multibody Dynamics Simulation Program Designs

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Equations of Motion in Acceleration Form

Most multibody dynamics simulation programs are designed  to solve the equations of motion for the system accelerations (acc = d/dt{v}). The equations are in the following form.

(Eq. 1)     J(x) d/dt{v} = f + G(x,v)

      where

J(x) = system mass matrix 
x = system states (hinge rates, and flexible body deformation coordinates)
v = generalized velocities (d/dt{x})
f = control and environmental forces/torque
G(x,v) = nonlinear forces

At each integration step, the solved accelerations are twice integrated to yield the system velocity and the coordinates.

Equations of Motion in Momentum Form

There are also programs that solve the equations of motion in the momentum form, which are written as follows::

(Eq. 2)    d/dt{H} = f + G'(x,v)

                where

G' = G + d/dt{J(x)}{v}
H = J(x){v}   (momentum equation)

In this design, Equation 2 and the current v are integrated to yield the generalized momentum vector, H, and the coordinates, {x}. At each integration step, the momentum equation is solved to obtain the generalized velocity, v. Thus, the system coordinates, {x}, and its velocities, {v}, are obtained at each step of the integration process.

Constraints

If the control design or the mechanism restriction calls for constraint handling, then Lagrange multiplier procedures are added to augment Eq. 1 and 2 to solve for constraint forces/torque to satisfy the specified restrictions.

 

Solution Methods

At each integration step, the simulation process requires solving a system of linear equations of the form J(x){a} = {b}. For Eq. 1 one would find d/dt{v} by

(Eq.3)     d/dt{v}= J(x)-1 { f + G(x,v)}

For Eq. 2, one needs to solve for {v} using the momentum equation by

(Eq.4)     {v}= J(x)-1H

The order(N3) methods are those that computes the system mass matrix J(x) given the current system states {x}, and then executes Equations 3 and 4. Here N denotes the degree of freedom of the mechanical system which is the row/column dimension of J(x). One can also use the more efficient order(N) method on these two equations. Published papers have shown that this method does not require the explicit computation of J(x) but nevertheless uses the recursive nature of the kinetics and kinematic relations between connected bodies to compute the system accelerations.

 

What is the Dynawiz Design?

Dynawiz is designed for spacecraft, mechanisms, and robotic systems. Its equations of motion are in the momentum and acceleration forms for the spacecraft package. The motion equations are in the acceleration form only for mechanisms and robotics packages. The solution method in each such package can be an order(N3) or order(N) method.

The XAL is for all mechanisms (rigid body only). Its equations of motion is generated by AUTOLEV(c) and they are in acceleration form. The equation formulation in AUTOLEV is based on the Kane's Method. The solution method used in XAL is an order(N3) method. See the XAL page for more details on what it has to offer.

The next few sections provide further details on the Dynawiz design.

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Last modified: Wednesday July 23, 2003.