diff --git a/paper/paper.org b/paper/paper.org index b3bd40a..31c370d 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -86,7 +86,7 @@ The Matlab code that was use to obtain the results are available in cite:dehaeze <> ** Model of a Rotating Positioning Platform # Introduce the fact that we need a simple system representing the rotating aspect -In order to study how the rotation of a positioning platforms does affect the use of force feedback, a simple model of an X-Y positioning stage on top of a rotating stage is developed. +In order to study how the rotation of a positioning platforms does affect the use of integral force feedback, a model of an XY positioning stage on top of a rotating stage is developed. # Simplest system where gyroscopic forces can be studied The model is schematically represented in Figure ref:fig:system and forms the simplest system where gyroscopic forces can be studied. @@ -94,20 +94,12 @@ The model is schematically represented in Figure ref:fig:system and forms the si The rotating stage is supposed to be ideal, meaning it induces a perfect rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\second}$. # X-Y Stage -The parallel X-Y positioning stage consists of two orthogonal actuators represented by three elements in parallel: -- a spring with a stiffness $k$ in $\si{\newton\per\meter}$ -- a dashpot with a damping coefficient $c$ in $\si{\newton\per\meter\second}$ -- an ideal force source $F_u, F_v$ - -# Payload -A payload with a mass $m$ in $\si{\kilo\gram}$ is mounted on the rotating X-Y stage. +The parallel XY positioning stage consists of two orthogonal actuators represented by three elements in parallel: a spring with a stiffness $k$ in $\si{\newton\per\meter}$, a dashpot with a damping coefficient $c$ in $\si{\newton\per\meter\second}$ and an ideal force source $F_u, F_v$. +A payload with a mass $m$ in $\si{\kilo\gram}$ is mounted on the (rotating) XY stage. # Explain the frames (inertial frame x,y, rotating frame u,v) -Two reference frames are used: -- an inertial frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ -- a uniform rotating frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ rigidly fixed on top of the rotating stage. $\vec{i}_w$ is aligned with the rotation axis - -The position of the payload is represented by $(d_u, d_v)$ expressed in the rotating frame. +Two reference frames are used: an inertial frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ and a uniform rotating frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ rigidly fixed on top of the rotating stage with $\vec{i}_w$ aligned with the rotation axis. +The position of the payload is represented by $(d_u, d_v, 0)$ expressed in the rotating frame. #+name: fig:system #+caption: Schematic of the studied System @@ -116,14 +108,14 @@ The position of the payload is represented by $(d_u, d_v)$ expressed in the rota ** Equations of Motion To obtain of equation of motion for the system represented in Figure ref:fig:system, the Lagrangian equations are used: -#+name: eq:lagrangian_equations +#+NAME: eq:lagrangian_equations \begin{equation} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i \end{equation} with $L = T - V$ the Lagrangian, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. -The constant rotation in the $(\vec{i}_x, \vec{i}_y)$ plane is here disregarded as it is imposed by the rotating stage. -#+name: eq:energy_functions_lagrange +The constant rotation in the $(\vec{i}_x, \vec{i}_y)$ plane is here disregarded as it is imposed by the ideal rotating stage. +#+NAME: eq:energy_functions_lagrange \begin{subequations} \begin{align} T & = \frac{1}{2} m \left( \left( \dot{d}_u - \Omega d_v \right)^2 + \left( \dot{d}_v + \Omega d_u \right)^2 \right) \\ @@ -133,8 +125,8 @@ The constant rotation in the $(\vec{i}_x, \vec{i}_y)$ plane is here disregarded \end{align} \end{subequations} -Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangian_equations gives the two coupled differential equations: -#+name: eq:eom_coupled +Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangian_equations gives two coupled differential equations +#+NAME: eq:eom_coupled \begin{subequations} \begin{align} m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \\ @@ -143,12 +135,12 @@ Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangi \end{subequations} # Explain Gyroscopic effects -The constant rotation of the system induces two Gyroscopic effects: -- Centrifugal forces: that can been seen as added negative stiffness along $\vec{i}_u$ and $\vec{i}_v$ +The uniform rotation of the system induces two Gyroscopic effects as shown in Eq. eqref:eq:eom_coupled: +- Centrifugal forces: that can been seen as added negative stiffness $- m \Omega^2$ along $\vec{i}_u$ and $\vec{i}_v$ - Coriolis Forces: that couples the motion in the two orthogonal directions One can verify that without rotation ($\Omega = 0$) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems: -#+name: eq:oem_no_rotation +#+NAME: eq:oem_no_rotation \begin{subequations} \begin{align} m \ddot{d}_u + c \dot{d}_u + k d_u &= F_u \\ @@ -168,11 +160,15 @@ To study the dynamics of the system, the differential equations of motions eqref \end{align} # Change of variables -To simply the analysis, the following change of variable is performed: -- $\omega_0 = \sqrt{\frac{k}{m}}$: Undamped natural frequency of the mass-spring system in $\si{\radian/\s}$ -- $\xi = \frac{c}{2 \sqrt{k m}}$: Damping ratio +To simplify the analysis, the undamped natural frequency $\omega_0$ and the damping ratio $\xi$ are used +\begin{subequations} + \begin{align} + \omega_0 &= \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second} \\ + \xi &= \frac{c}{2 \sqrt{k m}} + \end{align} +\end{subequations} -The transfer function matrix eqref:eq:Gd_m_k_c becomes equal to +The transfer function matrix $\bm{G}_d$ eqref:eq:Gd_m_k_c becomes equal to #+name: eq:Gd_w0_xi_k \begin{equation} \bm{G}_{d} = @@ -210,7 +206,7 @@ As the rotational speed increases, $p_{+}$ goes to higher frequencies and $p_{-} The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ is positive. Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forces exceeds the spring stiffness $k$. -In the rest of this study, rotational speeds smaller than the undamped natural frequency of the system are used ($\Omega < \omega_0$). +In the rest of this study, rotational speeds smaller than the undamped natural frequency of the system are assumed ($\Omega < \omega_0$). #+name: fig:campbell_diagram #+caption: Campbell Diagram : Evolution of the complex and real parts of the system's poles as a function of the rotational speed $\Omega$