Equations of Multi-Body Motion

Multi-body motion describes the dynamic behaviour of multiple interconnected rigid bodies. The rigid bodies are coupled using different types of joint and can translate and rotate relative to each other. The dynamic behaviour of the bodies is described by the equations of motion of the bodies and the constraint equations at the joints.

Two or more bodies are connected by one or more joints. A joint is defined by certain kinematic constraints. The position of the bodies relative to each other is therefore restricted by the kinematic constraint condition. The types of connection between bodies are described in the section Body Connections.

The general equation that describes the dynamic behaviour of a multi-body system is:

Figure 1. EQUATION_DISPLAY

M \ddot{q} = f

(4932)

where

q

is the vector of generalized coordinates,

f

is a generalized force, and

M

is the block-diagonal matrix of the inertia matrices of the rigid bodies:

Figure 2. EQUATION_DISPLAY

M = (\begin{matrix} M_{1} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & M_{n} \end{matrix})

(4933)

The equation of motion is combined with a constraint condition that restricts the kinematic degrees of freedom of the bodies:

Figure 3. EQUATION_TITLE

ϕ (q, t) = 0

(4934)

A constraint is expressed as a linear condition on the accelerations of the bodies. Forming the second derivative of Eqn. (4934) yields:

Figure 4. EQUATION_TITLE

J \ddot{q} = Q

(4935)

where $J$ is the Jacobian matrix of $ϕ$ , $\ddot{q}$ is the acceleration of the bodies, and $Q$ is an inhomogeneity.

To enforce the acceleration conditions of the constraints, a constraint force is added to the system. By introducing the Lagrangian multiplier $λ$ of all constraints, the workless constraint force is given by:

Figure 5. EQUATION_TITLE

f^{c} = J^{T} \cdot λ

(4936)

A vector $λ$ is sought such that the constraint force $f^{c}$ in combination with any external force $f^{e x t}$ such as gravity produces a motion that satisfies the constraints as given by Eqn. (4935).

With Eqn. (4936), it is possible to write Eqn. (4932) as:

Figure 6. EQUATION_TITLE

M \ddot{q} = J^{T} λ + f^{e x t}

(4937)

Solving for

\ddot{q}

Figure 7. EQUATION_TITLE

\ddot{q} = {M^{- 1} J}^{T} λ + M^{- 1} f^{e x t}

(4938)

yields the following linear system of equations:

Figure 8. EQUATION_TITLE

A λ = b

(4939)

where:

Figure 9. EQUATION_TITLE

A = J M^{- 1} J^{T}

(4940)

and

Figure 10. EQUATION_TITLE

b = - J M^{- 1} f^{e x t} + Q

(4941)

$λ$ is obtained from Eqn. (4939). With $λ$ known, Eqn. (4938) is integrated twice to yield the generalized coordinates vector.

Constraint Stabilization

The integration to obtain the coordinates vector $q$ is performed numerically. In Simcenter STAR-CCM+, the constraints are imposed in an acceleration form as given by Eqn. (4935). For such a method, it is natural that some constraint drift occurs due to round-off and numerical integration errors. The goal of constraint stabilization is to ensure that this drift does not accumulate over time and that it remains bounded.

Following Baumgarte [949], constraint stabilization is applied by modifying Eqn. (4935):

Figure 11. EQUATION_TITLE

J \ddot{q} = Q^{'}

(4942)

with:

Figure 12. EQUATION_TITLE

Q^{'} = Q - 2 α (J \dot{q} + \frac{\partial Φ}{\partial t}) - β^{2} Φ

(4943)

where $α$ and $β$ are parameters of the stabilization method.