Equations of Constrained Motions

Simcenter STAR-CCM+ provides motion options that apply physical constraints to the motion of the body, for example, forcing the body to move on a specified plane. These constraints allow certain degrees of freedom and restrict others.

This section describes the general form of the equations of motion in presence of physical constraints. Separate sections describe the specific form of the quantities that appear in these equations for the constrained motions available in Simcenter STAR-CCM+ (4-DOF Maneuvering Motion and Planar Motion Carriage).

The equations for constrained motions are derived from the general holonomic constraint equations for a rigid body:

Figure 1. EQUATION_DISPLAY

ϕ (d, t) = 0

(4900)

where $t$ is time and $d$ is the coordinate vector:

Figure 2. EQUATION_DISPLAY

d = (\begin{matrix} r_{x} \\ r_{y} \\ r_{z} \\ ϕ \\ θ \\ ψ \end{matrix})

(4901)

For a constrained body, not all components of $d$ are independent. It is therefore convenient to write Eqn. (4900) in terms of generalized independent coordinates. The minimum number of independent variables that are required to describe the motion of the rigid body is given by the number of degrees of freedom for the constrained body:

Figure 3. EQUATION_DISPLAY

f = 6 - p

(4902)

where

p

is the total number of constraints. The vector of independent coordinates can then be written as:

Figure 4. EQUATION_DISPLAY

q = (\begin{matrix} q_{1} \\ q_{2} \\ ⋮ \\ q_{f} \end{matrix})

(4903)

When using generalized coordinates, only $f$ equations are required. To derive the general equations for constrained motions, the body position (dependent coordinates) $r^{g}$ and the orientation matrix $T^{g}$ in the laboratory coordinate system are written in terms of the independent coordinates:

Figure 5. EQUATION_DISPLAY

\begin{array}{l} r^{g} = r (q, t) \\ T^{g} = T (q, t) \end{array}

(4904)

Differentiating these equations gives the translational and rotational velocities as:

Figure 6. EQUATION_DISPLAY

\begin{array}{l} v^{g} = \frac{\partial}{\partial q} r \dot{q} + \frac{\partial}{\partial t} r = J_{T} \dot{q} + v' \\ ω^{g} = \frac{\partial}{\partial q} t \dot{q} + \frac{\partial}{\partial t} t = J_{R} \dot{q} + ω' \end{array}

(4905)

where $J_{T} = \partial r / \partial q$ and $J_{R} = \partial t / \partial q$ are the $3 \times f$ Jacobian matrices for translation and rotation, with $\partial t = {(\begin{matrix} \partial t_{x} & \partial t_{y} & \partial t_{z} \end{matrix})}^{T}$ being an instant vector about which the rigid body rotates.

Differentiating Eqn. (4905) yields the accelerations of the rigid body:

Figure 7. EQUATION_DISPLAY

\begin{array}{l} {\dot{v}}^{g} = J_{T} \ddot{q} + L_{T} + \dot{v}' \\ {\dot{ω}}^{g} = J_{R} \ddot{q} + L_{R} + \dot{ω}' \end{array}

(4906)

where:

Figure 8. EQUATION_DISPLAY

\dot{v'} = \frac{\partial^{2} r}{\partial t^{2}}

(4907)

and

Figure 9. EQUATION_DISPLAY

\begin{array}{l} L_{T} = \frac{\partial}{\partial q} (J_{T} \dot{q}) + 2 \frac{\partial}{\partial q} v' \\ L_{R} = \frac{\partial}{\partial q} (J_{R} \dot{q}) + 2 \frac{\partial}{\partial q} v' + \frac{\partial}{\partial t} J_{R} \end{array}

(4908)

The governing equation under constraint conditions can then be written as:

Figure 10. EQUATION_DISPLAY

\overline{M} \ddot{q} + \overline{N} = F^{g}

(4909)

with

Figure 11. EQUATION_DISPLAY

\bar{M} = (\begin{matrix} m J_{T} \\ M^{g} J_{R} \end{matrix})

(4910)

and

Figure 12. EQUATION_DISPLAY

\overline{N} = (\begin{matrix} m \cdot L_{T} \overset{\cdot}{q} + m \cdot \dot{v}' + m [(J_{R} \ddot{q} + L_{R} \overset{\cdot}{q} + ω') \times r_{P C} + (J_{R} \overset{\cdot}{q} + ω') \times ((J_{R} \overset{\cdot}{q} + ω') \times r_{P C})] \\ M^{g} \cdot L_{R} \overset{\cdot}{q} \cdot M^{g} \cdot ω' + (J_{R} \overset{\cdot}{q} + ω') \times M^{g} \cdot (J_{R} \overset{\cdot}{q} + ω') \end{matrix})

(4911)

where $m$ is the mass of the body, $M^{g}$ is the tensor of the moments of inertia in the laboratory coordinate system, and $r_{P C}$ is a position vector that describes the distance between the center of rotation and the center of mass. Only for the planar motion carriage mechanism $r_{P C} \neq 0$ . For all other motions, $r_{P C} = 0$ .

The vector of forces and moments $F^{g}$ is defined as:

Figure 13. EQUATION_DISPLAY

F^{g} = (\begin{matrix} f^{g} \\ n^{g} \end{matrix})

(4912)

$F^{g}$ can be split into two parts:

Figure 14. EQUATION_DISPLAY

F^{g} = F^{i, g} + F^{q, g}

(4913)

$F^{i, g}$ stands for the impressed forces (or moments) acting on the body. These forces can either be obtained from the fluid flow or any external sources. $F^{q, g}$ stands for the forces (or moments) which maintain the given constraints.

$F^{q, g}$ can be eliminated by multiplying $J^{T}$ to both sides of Eqn. (4909), substituting Eqn. (4913), and by using $J^{T} F^{q} = 0$ . As a result, you get:

Figure 15. EQUATION_DISPLAY

(J^{T} \overline{M}) \ddot{q} + J^{T} \overline{N} = J^{T} F^{i, g}

(4914)

This equation has $f = 6 - p$ variables.