CES Mechanics II Lecture 15
Topics:\require{boldsymbol}
- Rigid body kinematics (continued) 3.1.1, 3.1.2, 3.1.3
- General plane motion.
When the rigid body undergoes general plane motion, different points A and B in the body have different velocities and accelerations, which can be related through relative velocity and relative acceleration:
\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{B/A},
\mathbf{a}_B = \mathbf{a}_A + \mathbf{a}_{B/A}.
When the body undergoes translation only, the relative position of two points remains constant:
\mathbf{r}_{B/A} = \mathrm{const}, \mathbf{v}_{B/A} = \mathbf{a}_{B/A} = \mathbf{0}.
But in general motion, all we can say is that the magnitude of the relative position vector remains constant:
|\mathbf{r}_{B/A}| = \mathrm{const},
and point B moves in a circular path around point A. Using again our knowledge of circular motion, we can write:
v_{B/A} = r_{B/A} \omega,
a_{{B/A}_n} = r_{B/A} \omega^2 = \frac{v_{B/A}^2}{R} = v_{B/A} \omega.
a_{{B/A}_t} = r_{B/A} \alpha,
or in vector notation:
\mathbf{v}_{B/A} = \boldsymbol{\omega} \times \mathbf{r}_{B/A},
\mathbf{a}_{{B/A}_n} = \boldsymbol{\omega} \times \mathbf{v}_{B/A} = \boldsymbol{\omega} \times \left( \boldsymbol{\omega} \times \mathbf{r}_{B/A} \right),
\mathbf{a}_{{B/A}_t} = \boldsymbol{\alpha} \times \mathbf{r}_{B/A}.
- Method of relative velocity.
In the method of relative velocity (relative Geschwindigkeit, Geschwindikeitsplan), we simply apply above equations to relate the velocity of two points A and B:
\mathbf{v}_B = \mathbf{v}_A + \boldsymbol{\omega} \times \mathbf{r}_{B/A},
resulting in 2 (planar motion) or 3 (general motion) kinematic equations. Presumably, enough information is already available (e.g., velocity of one of the points, direction of the velocities, angular velocity of the rigid body) that we can use these equations to obtain the missing information.
This equation can be also be considered as superposition of translation (\mathbf{v}_B = \mathbf{v}_A) with rotation about an axis at A.
- Method of instant center for velocity. 3.1.4
If we find a point O in the rigid body where the velocity is zero, the method of relative velocity simplifies, since all other points A rotate around that point as it were an axis:
\mathbf{v}_A = \boldsymbol{\omega} \times \mathbf{r}_{A/O}.
Such a point O is called the instant center for velocity (IC, Momentanpol) and is usually easy to identify in the planar motion: find two non-parallel known directions of velocities in the rigid body, draw two lines perpendicular to those velocities, and look for the intersection of those lines. Note that the IC may lay outside the rigid body itself, and generally does not stay fixed except in the case of rotation around a fixed axis. IC is also possible to find if we know two non-equal but parallel velocity vectors through interpolation or extrapolation—see ConceptTest below.
- Method of relative acceleration.
The method of relative acceleration (relative Beschleunigung) is analogous to the method of relative velocity: we apply the previously-derived equations to relate the acceleration of two points A and B:
\mathbf{a}_B = \mathbf{a}_A + \boldsymbol{\omega} \times \left( \boldsymbol{\omega} \times \mathbf{r}_{B/A} \right) + \boldsymbol{\alpha} \times \mathbf{r}_{B/A}.
There is no analogue to the IC for velocity in this case; the point of zero acceleration is usually not very easy to identify using graphical methods.
- General plane motion.
Corresponding Gross et al. Kinetik chapters are shown in blue.
Key techniques:
- Making sense of relative velocities and accelerations between two rigidly connected points.
- Identifying points on the rigid body for which some velocity data exists.
- Finding instant center for velocity given two known velocity vectors on the rigid body.
- Analyzing rigid body velocities using pure angular motion around the instant center.
ConcepTests (RWTH only):
