CES Mechanics II Lecture 19
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- Solving equations of plane rigid body motion (continued):
- Case 2: Rotation about fixed axis
If the axis of rotation is the central axis, then:
\bar{\mathbf{a}} = \mathbf{0}, and:
\mathbf{M}_G = \bar{I} \boldsymbol{\alpha}.
If the axis A is a non-central axis, then the most convenient form of the Euler's second law is:
\mathbf{M}_A = I_A \boldsymbol{\alpha},
and the acceleration of the center of mass \bar{\mathbf{a}} can be computed using the method of relative acceleration, using \mathbf{a}_A = \mathbf{0} as the reference acceleration.
- Case 3: General motion
Use either first or second form of the Euler's second law, but not the third.
- Case 2: Rotation about fixed axis
- Motion analysis.
Just like for particles and systems of particles, knowing instantaneous accelerations is often not enough. These often have to be integrated over time to obtain the full history of the motion. Since these accelerations often depend on current position, this leads to a system of ordinary differential equations in time. Solving such equations is beyond the scope of this course. Some useful results can be provided in a much simpler way using integral methods—work-energy and impulse-momentum principles—introduced in the next lectures.
Corresponding Gross et al. Kinetik chapters are shown in blue.
Key techniques:
- Understanding Euler's laws of planar motion of the rigid body.
- Choosing a coordinate system orientation and moment balance equation which simplify the equations of motion
- Identifying the unknowns in the equations of motion
- applying kinematic constraints and method of relative acceleration to eliminate extra unknowns
ConcepTests (RWTH only):
