Planar Motion Carriage

The motion of a body that is forced to move on a plane has three degrees of freedom.

For planar motion, the independent coordinate vector can be written as:

Figure 1. EQUATION_DISPLAY

q = (\begin{matrix} z \\ θ \\ ϕ \end{matrix})

(4921)

For the planar motion carriage, the Jacobian matrices and other quantities in Eqn. (4909) reduce to:

Figure 2. EQUATION_DISPLAY

J_{T} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix})

(4922)

Figure 3. EQUATION_DISPLAY

J_{R} = (\begin{matrix} 0 & - \sin ψ & \cos ψ \cdot \cos θ \\ 0 & \cos ψ & \sin ψ \cdot \cos θ \\ 0 & 0 & - \sin θ \end{matrix})

(4923)

Figure 4. EQUATION_DISPLAY

\begin{array}{l} L_{T} = 0 \\ L_{R} = (\begin{matrix} 0 & - \cos ψ \frac{\partial ψ}{\partial t} - \cos ψ \cdot \sin θ \cdot \dot{ϕ} & - \sin ψ \cdot \cos θ \cdot \frac{\partial ψ}{\partial t} \\ 0 & - \sin ψ \frac{\partial ψ}{\partial t} - \sin ψ \cdot \sin θ \cdot \dot{ϕ} & \cos ψ \cdot \cos θ \cdot \frac{\partial ψ}{\partial t} \\ 0 & - \cos θ \cdot \dot{ϕ} & 0 \end{matrix}) \\ v' = (\begin{matrix} \frac{\partial r_{x}}{\partial t} \\ \frac{\partial r_{y}}{\partial t} \\ 0 \end{matrix}) \\ ω' = (\begin{matrix} 0 \\ 0 \\ \frac{\partial ψ}{\partial t} \end{matrix}) \\ \dot{v}' = (\begin{matrix} \frac{\partial^{2} r_{x}}{\partial t^{2}} \\ \frac{\partial^{2} r_{y}}{\partial t^{2}} \\ 0 \end{matrix}) \\ \dot{ω}' = (\begin{matrix} 0 \\ 0 \\ \frac{\partial^{2} ψ}{\partial t^{2}} \end{matrix}) \end{array}

(4924)

Simcenter STAR-CCM+ provides various methods for modeling planar motion. The position vector and the orientation matrix depend on the method in use.

Planar Motion Mechanism (PMM)

The Planar Motion Mechanism is a method of the Planar Motion Carriage that prescribes the motion in the X-Y plane in the form of a sinusoidal sway motion. With this method, the body position $r$ can be written as:

Figure 5. EQUATION_DISPLAY

r = (\begin{matrix} r_{x} \\ r_{y} \\ r_{z} \end{matrix}) = (\begin{matrix} V_{0} \cdot t \\ y_{0} \cdot \sin (2 π f \cdot t) \\ z \end{matrix})

(4925)

where $V_{0}$ is the forward velocity of the body, $y_{0}$ is the sway amplitude, and $f′$ is the sway frequency.

The orientation $T$ is given by Eqn. (4917), with the roll angle $ϕ = 0$ .

If pure yawing is activated, the yawing angle is given by:

Figure 6. EQUATION_DISPLAY

ψ = ψ_{0} \cdot \cos (2 π f \cdot t)

(4926)

with

Figure 7. EQUATION_DISPLAY

ψ_{0} = \arctan (\frac{y_{0} \cdot 2 π f}{V_{0}})

(4927)

else $ψ = 0$ .

Rotating Arm Motion

The Rotating Arm Motion mechanism is a method of the Planar Motion Carriage that prescribes the motion in the X-Y plane in the form of a rotation. The result is a circular path of the body in the X-Y plane. With this method, the body position $r$ can be written as:

Figure 8. EQUATION_DISPLAY

r = (\begin{matrix} r_{x} \\ r_{y} \\ r_{z} \end{matrix}) = (\begin{matrix} r_{c, x} + (r_{0, x} - r_{c, x}) \cos ω t - (r_{0, y} - r_{c, y}) \sin ω t \\ r_{c, y} + (r_{0, x} - r_{c, x}) \sin ω t - (r_{0, y} + r_{c, y}) \cos ω t \\ z \end{matrix})

(4928)

where $\begin{matrix} r_{c, x} \end{matrix}, \begin{matrix} r_{c, y} \end{matrix}, \begin{matrix} r_{c, z} \end{matrix}$ define the center of rotation of the rotating arm, and $\begin{matrix} r_{0, x} \end{matrix}, \begin{matrix} r_{0, y} \end{matrix}, \begin{matrix} r_{0, z} \end{matrix}$ define the initial body position, that is, the location where the body is initially attached to the rotating arm.

The yaw angle is given by:

Figure 9. EQUATION_DISPLAY

ψ = ω \cdot t

(4929)

General Planar Motion

The General Planar Motion mechanism is a method of the Planar Motion Carriage that prescribes the motion in the X-Y plane in the form of a user-defined trajectory. With this method, the body position $r$ can be written as:

Figure 10. EQUATION_DISPLAY

r = (\begin{matrix} r_{x} \\ r_{y} \\ r_{z} \end{matrix}) = (\begin{matrix} x (t) \\ y (t) \\ z \end{matrix})

(4930)

The yaw angle is given by:

Figure 11. EQUATION_DISPLAY

ψ = ψ (t)

(4931)

where $x, y, ψ$ are user-defined functions of time.