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CES Mechanics II Lecture 14

RWTH Aachen University, Bachelor program in Computational Engineering Science

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Topics (rigid body kinematics):\require{boldsymbol}

• Rigid body kinematics 3.1.1, 3.1.2, 3.1.3

A rigid body (Festkörper) is a collection of points (particles) such that distance between any two points remains constant (no extension of compression). This property is enough to show that an angle between any two lines in a rigid body remains constant.

• Plane angular motion.

We will consider for now only planar motion (ebene Bewegung), which can be described purely in 2D coordinate system. For that, we will assume that the motion of the (3D) rigid body can be entirely described by the motion of its selected cross-section in the plane of motion, with all other points moving parallel to that plane at fixed distances.

A rigid body can undergo:

• translation (like particles): any line remains parallel, motion of one point determines motion of all other points,
• rotation about fixed axis: one line fixed in space, other points move in circles around it,
• general plane motion:superposition of translation and rotation, e.g., wheel.

Consider a line AB embedded in the rigid body in the plane of motion. Because of the property that angles between lines remain constant, it is sufficient to describe the change in angle of that line, since all other lines move the same way. Between time t and t + \Delta t, the line (and the rigid body) undergoes the angular displacement (Drehwinkel) \Delta {\theta} = \theta(t + \Delta t) - \theta(t).

The angular velocity (Winkelgeschwindigkeit) \omega is the rate of angular displacement over time:

\omega = \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d \theta}{d t} = \dot{\theta}.

This is the angular velocity of all lines, and thus the angular velocity of the rigid body.

The angular acceleration (Winkelbeschleunigung) \alpha follows:

\alpha = \frac{d \omega}{d t} = \dot{\omega} = \frac{d^2 \theta}{d t^2} = \ddot{\theta}.

This is the angular acceleration of all lines, and thus the angular acceleration of the rigid body.

Another way of computing the angular acceleration is:

\alpha = \frac{d \omega}{d t} = \frac{d \omega}{d \theta} \frac{d \theta}{d t} = \omega \frac{d \omega}{d t}.

These concepts are analogous to linear displacement \Delta x, velocity v and acceleration a. The dimensions of angular displacement, velocity and acceleration are radians, radians per second and radians per second squared, respectively.

In 3D general motion, vector angular displacements \Delta \boldsymbol{\theta}, velocity \boldsymbol{\omega} and acceleration \boldsymbol{\alpha} are used. These are also useful in describing planar motion; however, \boldsymbol{\omega} and \boldsymbol{\alpha} are always perpendicular to the plane of motion in that case (out of the plane for counterclockwise rotation and into the plane for clockwise rotation).

Note that a finite angular displacement in 3D \Delta \boldsymbol{\theta} cannot be considered a vector in mathematical sense, since addition of two angular displacement vectors is not commutative (order of applying the displacement matters). Its infinitesimal version d \boldsymbol{\theta} does not present such difficulties, and therefore angular velocity and acceleration can be considered vectors.

• Rotation around fixed axis.

In this case, every point in the rigid body undergoes the rotation about the axis. From kinematics of particles, we know that in this case (using polar coordinates centered at the axis):

v = R \dot{\theta},

a_n = R \dot{\theta}^2 = v \dot{\theta} = \frac{v^2}{R},

a_t = R \ddot{\theta},

where R is the distance from the axis to the point where velocity and acceleration are being determined. Substituting the angular velocity and acceleration (the same for all points in the rigid body):

v = R \omega,

a_n = R \omega^2 = v \omega = \frac{v^2}{R},

a_t = R \alpha,

or, using vector notation:

\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r},

\mathbf{a}_n = \boldsymbol{\omega} \times \mathbf{v} = \boldsymbol{\omega} \times \left( \boldsymbol{\omega} \times \mathbf{r} \right),

\mathbf{a}_t = \boldsymbol{\alpha} \times \mathbf{r},

where \mathbf{r} is the vector from arbitrary point on the axis of rotation to the point in the rigid body where velocity and acceleration are being determined.

Corresponding Gross et al. Kinetik chapters are shown in blue.

Key techniques:

• Describing pure rotation in a plane.
• Relating point-wise velocity and acceleration to overall angular velocity and acceleration.

ConcepTests (RWTH only):

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