Rigid Body Kinematics
Rigid bodies
Complete in "Rigid bodies"
A rigid body is an extended area of material that includes all the points inside it, and which moves so that the distances and angles between all its points remain constant. The location of a rigid body can be described by the position of one point $P$ inside it, together with the rotation angle of the body (one angle in 2D, three angles in 3D).| location description | velocity description | |
|---|---|---|
| point mass | position vector \( \vec{r}_P \) | velocity vector \( \vec{v}_P \) |
| rigid body in 2D |
position vector \( \vec{r}_P \) angle $\theta$ |
velocity vector \( \vec{v}_P \) angular velocity \( \omega \) |
| rigid body in 3D |
position vector \( \vec{r}_P \) angles \( \theta,\phi,\psi \) |
velocity vector \( \vec{v}_P \) angular velocity vector \( \vec{\omega} \) |
Neither point masses nor rigid bodies can physically exist, as no body can really be a single point with no extent, and no extended body can be exactly rigid. Despite this, these are very useful models for mechanics and dynamics.
Rotating rigid bodies
Complete in "Rigid bodies"
All points on a rigid body have the same angular rotation angles, as we can see on the figure below. Because the angular velocity is the derivative of the rotation angles, this means that every point on a rigid body has the same angular velocity \( \vec{\omega} \), and also the same angular acceleration \( \vec{\alpha} \).
In 2D the angle \( \theta \) of a rigid body the angle of rotation from a fixed reference (typically the \( \hat\imath \) direction), measured positive counter-clockwise. The angular velocity is \( \omega = \dot\theta \) and the angular acceleration is \( \alpha = \dot\omega = \ddot\theta \). The vector versions of these are \( \vec\omega = \omega \, \hat{k} \) and \( \vec\alpha = \alpha\,\hat{k} \), where \( \hat{k} \) is the out-of-plane direction.
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Rotations in 3D are significantly more complicated than rotations in 2D. Unlike positions, velocities, etc, which simply go from 2D vectors to 3D vectors, rotational quantities go from scalars in 2D to full 3D vectors in 3D. Angular velocity and angular acceleration are somewhat straightforward, so equations #rkg-er hold in both 2D and 3D, but understanding the rotations themselves is significantly more complicated. There are three main ways that 3D rotations can be represented:
Detailed study of rotations in 3D is necessary for a full understanding of topics ranging from satellite attitude control to articulated robot construction, and is usually covered in advanced dynamics courses.
Rigid body velocity
Complete in "Rigid bodies"
Constrained motion
Complete in "Constraints and constrained motions"
Rigid body acceleration
Complete in "Rigid bodies"
Gears
Include the information in Fig \ref fig:GearsRecap as a broad introduction to the topic
Standard sign convention
Add information shown in Fig \ref fig:gears
Include animation of examples as the ones in Fig \ref fig:GearsExamples
Accelerations in contact (no slip)
Add the information included in Fig \ref fig:GearsAcceleration
Applications
Train passenger comfort
This topics need to be clearly organized and expanded. Refer to Fig \ref fig:AppTrainPassenger
. Application for "Rigid body acceleration".
Steering geometry
Complete in "Steering geometry".
This refers to "Rigid Bodies".Knee Joint
Complete in "Four-Bar Linkages" under the subtitle "Example: Knee joint (constrained motion)".
This refers to "Constrained motion".Suspensions with Watt's linkage
Complete in "Four-Bar Linkages" under the subtitle "Example: Suspensions with Watt's linkage (constrained motion)".
This refers to "Constrained motion".Aerobie Orbiter
This topic was mentioned in lecture but it was not expanded.
Application for "Rigid body rotation".