Rotation of a system of bodies on an axis

Question

I am doing a physical simulation and got a question regarding simulation of rotational movement of the system of interconnected rods.

Let’s have two rods connected by the axis which pass neither through center of mass (CoM) of any of parts nor through the CoM of the system. These two rods rotate in opposite directions around this axis.

What forces/torques/etc. should be taken into account to model the movement of this system, namely to calculate the positions, orientations, velocities, etc. after some small time dt?

I assume that the system should rotate around its CoM. The question is what is the rotation axis direction? One of the points is CoM, what is the second point? I am sure that the axis shouldn’t be parallel the green rotation axis, since we have leverages here, but can’t figure out correct rotation.

What is the missing part? What kind of gut feeling I am lacking here?

The second question is, how the situation would change if instead of axis which allows constant rotation speeds for rods, I put a motor which will speed up the rotation? How the calculation would change?

Just to make sure that we are on the same page: nothing in this system is fixed to outer space; for example, the green axis also a part of the system that have all the necessary degrees of freedom.

Answer 1

Two connected bodies, the vector edition.

Consider two bodies connected together with a pin joint as seen below

Quantities with the subscripts 1, 2 related to either of the two bodies, and the subscript k relates to the joint connecting the two bodies.

Definitions

Each body (use i=1,2) has mass $m_i$ and mass moment of inertia ${\rm I}_i$ oriented along the inertial frame basis-vectors. I am tracking their center of mass with position $\vec{r}_i$, velocity $\vec{v}_i$ and acceleration $\vec{a}_i$. Also rotational velocity $\vec{\omega}_i$ and rotational acceleration $\vec{\alpha}_i$. Additionally, each might have an external force applied $\vec{W}_i$ (such as gravity).

The joint has its axis along the direction $\vec{z}_k$ and is located at $\vec{r}_k$. The reaction forces/moments of the joint $\vec{F}_k$ and $\vec{M}_k$ act in a positive fashion on body 2, and in equal and opposite sense on body 1.

Equations of motion

Applying the equations of motion to this system is somewhat straightforward. There are two equations for each body, one translational and one rotational.

$$\begin{aligned}\vec{W}_{1}-\vec{F}_{k} & =m_{1}\vec{a}_{1}\\ \vec{W}_{2}+\vec{F}_{k} & =m_{2}\vec{a}_{2}\\ -\vec{M}_{k}-\left(-\vec{c}_{1}\right)\times\vec{F}_{k} & ={\rm I_{1}\vec{\alpha}_{1}+\vec{\omega}_{1}\times{\rm I}_{1}\vec{\omega}_{1}}\\ \vec{M}_{k}+\left(-\vec{c}_{2}\right)\times\vec{F}_{k} & ={\rm I_{2}\vec{\alpha}_{2}+\vec{\omega}_{2}\times{\rm I}_{2}\vec{\omega}_{2}} \end{aligned} \tag{1}$$

_{I am using the relative locations of the COM to the joint to simplify the equations $$\begin{aligned}\vec{c}_{1} & =\vec{r}_{1}-\vec{r}_{k}\\ \vec{c}_{2} & =\vec{r}_{2}-\vec{r}_{k} \end{aligned}$$}

Velocity Kinematics

The velocities of the two bodies are related due to the pin joint

$$\begin{aligned}\vec{v}_{k} & =\vec{v}_{1}+\vec{\omega}_{1}\times\left(-\vec{c}_{1}\right)\\ \vec{v}_{2} & =\vec{v}_{k}+\vec{\omega}_{2}\times\left(\vec{c}_{2}\right)\\ \vec{\omega}_{2} & =\vec{\omega}_{1}+\vec{z}_{k}\dot{\theta}_{k} \end{aligned} \tag{2}$$

Here $\vec{v}_k$ is the translational velocity of the joint position in space. Notice that the angle $\theta_k$ between the bodies is defined in a relative sense such that the orientation of body 2 would be calculated from the orientation of body 1 in a sequence of rotations $$ {\rm R}_2 = {\rm R}_1 {\rm rot}(\vec{z}_k,\,\theta_k)$$

Acceleration kinematics

The direct derivative of (2) above gives us the accelerations of the two centers of mass $\vec{a}_1$ and $\vec{a}_2$ which is used in the equations of motion (1).

$$\begin{aligned}\vec{a}_{k} & =\vec{a}_{1}+\vec{\alpha}_{1}\times\left(-\vec{c}_{1}\right)+\vec{\omega}_{1}\times\left(\vec{\omega}_{1}\times\left(-\vec{c}_{1}\right)\right)\\ \vec{a}_{2} & =\vec{a}_{k}+\vec{\alpha}_{2}\times\left(\vec{c}_{2}\right)+\vec{\omega}_{2}\times\left(\vec{\omega}_{2}\times\vec{c}_{2}\right)\\ \vec{\alpha}_{2} & =\vec{\alpha}_{1}+\vec{z}_{k}\ddot{\theta}_{k}+\vec{\omega}_{2}\times\vec{z}_{k}\dot{\theta}_{k} \end{aligned} \tag{3}$$

Joint Condition

There is one more relationship needed to solve for the unknown quantities $\vec{\alpha}_1$, $\vec{\alpha}_2$, $\vec{a}_1$, $\vec{a}_2$, $\vec{a}_k$, $\vec{F}_k$, $\vec{M}_k$, $\ddot{\theta}_k$. So far we have 7 vector equations in (1) and (3) for 21 components. But we have 7 vector unknowns, and 1 scalar unknown (the relative angle acceleration).

Consider the power generated by the joint as $\mathcal{P} = \dot{\theta}_k \, \tau_k$ as it only rotates, and equate it to the vector form of the power $\mathcal{P} = \vec{M}_k \cdot ( \vec{\omega}_2 - \vec{\omega}_1)$ where $\cdot$ is the vector dot product

The missing equation is

$$ \tau_{k}=\vec{z}_{k}\cdot\vec{M}_{k} \tag{4} $$

Solution

Form a 22×22 system of equations from (1), (3) and (4) at each time step, in order to find and integrate the accelerations into the next time step.

$$\small \left[\begin{array}{ccc|cc|cc|c} m_{1} & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & m_{2} & 0 & 0 & 0 & -1 & 0 & 0\\ \hline 0 & 0 & 0 & {\rm I}_{1} & 0 & -\vec{c}_{1}\times & 1 & 0\\ 0 & 0 & 0 & 0 & {\rm I}_{2} & \vec{c}_{2}\times & -1 & 0\\ \hline 1 & 0 & -1 & \vec{c}_{1}\times & 0 & 0 & 0 & 0\\ 0 & 1 & -1 & 0 & \vec{c}_{2}\times & 0 & 0 & 0\\ \hline 0 & 0 & 0 & -1 & 1 & 0 & 0 & -\vec{z}_{k}\\ \hline 0 & 0 & 0 & 0 & 0 & 0 & \vec{z}_{k}^{\intercal} & 0 \end{array}\right]\left[\begin{array}{c} \vec{a}_{1}\\ \vec{a}_{2}\\ \vec{a}_{k}\\ \hline \vec{\alpha_{1}}\\ \vec{\alpha_{2}}\\ \hline \vec{F}_{k}\\ \vec{M}_{k}\\ \hline \ddot{\theta}_{k} \end{array}\right]=\left[\begin{array}{c} \vec{W}_{1}\\ \vec{W}_{2}\\ \hline -\vec{\omega}_{1}\times{\rm I}_{1}\vec{\omega}_{1}\\ -\vec{\omega}_{2}\times{\rm I}_{2}\vec{\omega}_{2}\\ \hline \vec{\omega}_{1}\times\left(\vec{\omega}_{1}\times\vec{c}_{1}\right)\\ \vec{\omega}_{2}\times\left(\vec{\omega}_{2}\times\vec{c}_{2}\right)\\ \hline \vec{\omega}_{2}\times\vec{z}_{k}\dot{\theta}_{k}\\ \hline \tau_{k} \end{array}\right] \tag{5} $$

_{Here a $1$ correspond to the 3×3 identity matrix, and $\vec{r}\times$ to the 3×3 skew symmetric matrix $$\vec{r}\times = \pmatrix{0 & -z & y\\ z & 0 & -x \\ -y & x & 0}$$ and $\vec{z}^\intercal$ is the transpose of the vector making it a row vector $\pmatrix{x & y & z}$ from a column vector.}

Answer 2

constraint equations translation \begin{align*} &\mathbf R_1+ \mathbf S_1\, \mathbf u_1-\left(\, \mathbf R_2+ \mathbf S_2\, \mathbf u_2\,\right)= \mathbf 0\tag 1 \end{align*} where $~ \mathbf S_i~$ are the transformation of the body fixed coordinate system to inertial system

equations rotation \begin{align*} \mathbf\omega_2=\mathbf\omega_1+ \mathbf n\,\dot\psi\tag 2 \end{align*} $~ \mathbf n~$ is the hinge axis

thus the generalized coordinate are the three positions of body 1 $~( \mathbf R_1)~$ three angular velocity $~\mathbf\omega_1~$ of body 1 ,plus the angular velocity ($~\dot\psi)$

with:

\begin{align*} &\mathbf{\dot{S}_i}= \left[\mathbf\omega\right]_\times\,\mathbf S_i\,\quad\Rightarrow\\ &\mathbf\omega_i= \mathbf J_i\, \mathbf{ \dot{\phi}_i}\quad i=1,2 \quad, \mathbf{ \dot{\phi}_i}= \mathbf{ J}_i^{-1}\mathbf\omega_i \end{align*}

from the above you can obtain the kinematic equations

\begin{align*} & \mathbf R_1= \mathbf R_1(t)\quad, \mathbf \omega_1= \mathbf \omega_1(t)\quad, \dot\psi=\dot\psi(t)\\ &\Rightarrow\\ &\mathbf\omega_2(t)=\mathbf\omega_1(t)+ \mathbf n\,\dot{\psi}(t)\\ &\mathbf{\dot{\phi}_i}=\mathbf{J}_i^{-1}\mathbf\omega_i(t)\\\\ &\text{from Eq. (1)}\\\\ &\mathbf R_2(t)=\mathbf R_1(t)+ \mathbf S_1(t)\, \mathbf u_1- \mathbf S_2(t)\, \mathbf u_2 \end{align*}

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Example 2D

\begin{align*} &\mathbf R_1=\begin{bmatrix} x(t) \\ y(t)\\ 0 \\ \end{bmatrix}\quad, \mathbf S_1=\left[ \begin {array}{ccc} \cos \left( \varphi _{{1}} \right) &-\sin \left( \varphi _{{1}} \right) &0\\ \sin \left( \varphi _{{1}} \right) &\cos \left( \varphi _{{1}} \right) &0 \\ 0&0&1\end {array} \right]\quad, \mathbf S_2=\left[ \begin {array}{ccc} \cos \left( \varphi _{{2}} \right) &-\sin \left( \varphi _{{2}} \right) &0\\ \sin \left( \varphi _{{2}} \right) &\cos \left( \varphi _{{2}} \right) &0 \\ 0&0&1\end {array} \right]\\ &\mathbf{u}_1=\left[ \begin {array}{c} u_{{{\it x1}}}\\ u_{{{\it y1}}}\\ 0\end {array} \right] \quad, \mathbf{u}_2= \left[ \begin {array}{c} u_{{{\it x2}}}\\ u_{{{\it y2}}}\\ 0\end {array} \right]\\ &\mathbf{\omega}_1=\left[ \begin {array}{c} 0\\ 0\\ \omega \left( t \right) \end {array} \right] \quad, \mathbf{n}=\begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix}\quad \dot{\varphi}_1=\omega(t)\quad, \varphi_1(t)=\int\omega(t)\,dt\\ \quad\Rightarrow\\ &\mathbf{\omega}_2= \left[ \begin {array}{c} 0\\ 0\\ \omega \left( t \right) +{\frac {d}{dt}}\psi \left( t \right) \end {array} \right] \quad, \dot{\varphi_2}=\omega(t)+\dot{\psi}(t)\quad,\varphi_2(t)=\int ...\,dt\\\\ &\mathbf{R}_2=\left[ \begin {array}{c} x \left( t \right) +\cos \left( \varphi _{{1 }} \right) u_{{{\it x1}}}-\sin \left( \varphi _{{1}} \right) u_{{{\it y1}}}-\cos \left( \varphi _{{2}} \right) u_{{{\it x2}}}+\sin \left( \varphi _{{2}} \right) u_{{{\it y2}}}\\ y \left( t \right) +\sin \left( \varphi _{{1}} \right) u_{{{\it x1}}}+\cos \left( \varphi _{{1}} \right) u_{{{\it y1}}}-\sin \left( \varphi _{{2 }} \right) u_{{{\it x2}}}-\cos \left( \varphi _{{2}} \right) u_{{{\it y2}}}\\ 0\end {array} \right]\\ &\mathbf v_2=\frac{d}{dt}\mathbf R_2(t)\quad, \mathbf a_2=\frac{d}{dt}\mathbf v_2(t) \end{align*}

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3D and 2D

assume that the hinge rotation axis is fixed with body 1

input

$~x(t)~,y(t)~,z(t)~,\phi_x(t)~,\phi_y(t)~,\phi_z(t),~\psi(t)$

output

$~\mathbf{R}_2(t)~,\mathbf{S}_2(t)$

the equatiuons are: \begin{align*} &\mathbf{R}_2=\mathbf{R}_1+\mathbf{S}_1\,\mathbf{u}_1-\mathbf{S}_2\,\mathbf{u}_2\quad,\text{where}\\ &\mathbf{S}_2=\mathbf{S}_1\,\mathbf{S}_h(\mathbf{n}~,\psi) \end{align*} the hinge transformation matrix $~\mathbf S_h~$ you can use the rodriguez matrix

the velocity $~\mathbf v_2~$ and angular velocity $~\mathbf \omega_2~$

\begin{align*} &\mathbf{v}_2=\mathbf{\dot{R}}_1+\mathbf{\dot{S}}_1\,\mathbf{u}_1- \mathbf{\dot{S}}_2\,\mathbf{u}_2\\ &\mathbf{\omega}_2=\mathbf{\omega}+\mathbf{ S}_1\,\mathbf{\omega}_h\quad,\text{where}\\\\ &[~\mathbf \omega]_\times=\mathbf{\dot{S}}_1\,\mathbf S_1^T\quad, [~\mathbf \omega_h]_\times=\mathbf{\dot{S}}_h\,\mathbf S_h^T \end{align*} Example \begin{align*} &\phi_x=3\,t~,\phi_y=t~,\phi_z=5\,t~,\psi(t)=10 t\\ &x=y=z=3\,t\quad,\dot{x}=\dot{y}=\dot{z}=3\\ &\mathbf n= \begin{bmatrix} 1 & 0 & 0 \\ \end{bmatrix}^T\quad,\Rightarrow\\ \\ &\mathbf S_1=\mathbf S_x(\phi_x)\,\mathbf S_y(\phi_y)\,\mathbf S_z(\phi_z)\quad,\mathbf{\dot{S}}_1=\frac{d}{dt}\,\mathbf S_1(t)\\\ &\mathbf S_h(\mathbf n,\psi)=\left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \psi \right) &-\sin \left( \psi \right) \\ 0&\sin \left( \psi \right) &\cos \left( \psi \right) \end {array} \right] \quad,\mathbf{\dot{S}}_h=\frac{d}{dt}\mathbf S_h(\mathbf n,\psi) \\\\ &\mathbf\omega= \left[ \begin {array}{c} 3+5\,\sin \left( t \right) \\ \cos \left( 3\,t \right) -5/2\,\sin \left( 2\,t \right) -5/2\,\sin \left( 4\,t \right) \\ \sin \left( 3\,t \right) +5/2\,\cos \left( 4\,t \right) +5/2\,\cos \left( 2\,t \right) \end {array} \right] \\ &\omega_h=\dot{\psi} \\\\ &\mathbf R_2(t)=\ldots\quad,\mathbf S_2(t)=\ldots\\ &\mathbf v_2(t)=\ldots\quad,\mathbf\omega_2(t)=\ldots \end{align*}

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\begin{align*} &\text{Rodriguez transformation matrix }\\ &\mathbf{S}_h= \mathbf{I}_3+\sin(\psi)\,[\mathbf{n}]_\times+ (1-\cos(\psi)) [\mathbf{n}]_\times\,[\mathbf{n}]_\times\\ &\mathbf{\omega}_h=\dot{\psi}\,\mathbf{n}\quad,\mathbf{n}\cdot\mathbf{n}=1 \end{align*}

how to obtain the angular velocity $~\mathbf\omega~$ from the rotation matrix $~\mathbf S~$ between the body and inertial system

\begin{align*} &\dot{\mathbf{S}}=[\mathbf\omega]_\times\,\mathbf{S}\quad\Rightarrow \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \\ \end{bmatrix} =\dot{\mathbf{S}}\,\mathbf{S}^T \\ &\textbf{Example}\\ &\mathbf{S}=\left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \phi \right) &-\sin \left( \phi \right) \\ 0&\sin \left( \phi \right) &\cos \left( \phi \right) \end {array} \right] \quad, \dot{\mathbf{S}}=\dot{\varphi} \left[ \begin {array}{ccc} 0&0&0\\ 0&-\sin \left( \phi \right) &-\cos \left( \phi \right) \\ 0&\cos \left( \phi \right) &-\sin \left( \phi \right) \end {array} \right]\quad\Rightarrow\\ &\begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \\ \end{bmatrix}= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -\dot{\varphi} \\ 0 & \dot{\varphi} & 0 \\ \end{bmatrix}\quad,\vec{\omega}=\begin{bmatrix} \dot{\varphi} \\ 0 \\ 0 \\ \end{bmatrix} \end{align*}