CES Mechanics II Lecture 1
RWTH Aachen University, Bachelor program in Computational Engineering Science
This page uses Javascript jsMath to display mathematical expressions. For best results, use a modern browser and download optimized fonts.
Topics:
- Statics review (see below).
- Outline of dynamics: We will proceed from a single particle (which has no capability to rotate) to system of particles, and then to rigid body. In each case, we will discuss first the kinematics, i.e., the motion without regard for its causes, and then kinetics, i.e., forces causing motion. In kinematics, we will talk about absolute motion (in fixed frame of reference) and then about relative motion (in moving frame of reference). In kinetics, we will consider first the standard force-mass-acceleration approach, based on Newton's second law, and then two integral methods, namely work-energy principle and impulse-momentum principle.
- Differentiation of vector functions:
- \frac{d \left( m \mathbf{A} \right)}{d u} = \frac{d m}{d u} \mathbf{A} + m \frac{d \mathbf{A}}{d u}
- \frac{d \left( \mathbf{A} + \mathbf{B} \right)}{d u} = \frac{d \mathbf{A}}{d u} + \frac{d \mathbf{B}}{d u}
- \frac{d \left( \mathbf{A} \cdot \mathbf{B} \right)}{d u} = \frac{d \mathbf{A}}{d u} \cdot \mathbf{B} + \mathbf{A} \cdot \frac{d \mathbf{B}}{d u}
- \frac{d \left( \mathbf{A} \times \mathbf{B} \right)}{d u} = \frac{d \mathbf{A}}{d u} \times \mathbf{B} + \mathbf{A} \times \frac{d \mathbf{B}}{d u}
- \left| \frac{d \mathbf{A}}{d u} \right| \neq \frac{ d | \mathbf{A} |}{d u}
- Particle kinematics 1.1
- Position Ortsvektor \mathbf{r}(t) and path coordinate Bahnkoordinate s(t) are now considered functions of time t.
- Velocity Geschwindigkeit is defined as
\mathbf{v}(t) = \frac{d \mathbf{r}(t)}{d t} = \dot{\mathbf{r}}(t)
and is a vector always tangent to the path.
- Speed Bahngeschwindigkeit is defined as
v(t) = \frac{d s(t)}{d t} = \dot{s}(t)
and is a scalar with magnitude equal to that of velocity.
- Acceleration Beschleunigung is defined as
\mathbf{a}(t) = \frac{d \mathbf{v}(t)}{d t} = \dot{\mathbf{v}}(t) = \ddot{\mathbf{r}}(t)
and is a vector not necessarily tangent to the path.
Corresponding Gross et al. Kinetik chapters are shown in blue.
Key techniques:
- differentiation of vector functions
ConcepTests (RWTH only):
Review of statics:
- equilibrium if \mathbf{R} = \mathbf{0}, \mathbf{C}^R = \mathbf{0} (3D, 2D)
- what if not? dynamics!
- how to compute \mathbf{R}, \mathbf{C}^R?
- forces can be added to form \mathbf{R} only for concurrent systems
- moving action line of a force generates a couple of transfer \mathbf{C}^T = \mathbf{r} \times \mathbf{F}
- couples of transfer can be added to any other couples to form \mathbf{C}^R
- \mathbf{C}^R can be often eliminated by moving \mathbf{R} to a point where \mathbf{C}^R = \mathbf{r} \times \mathbf{R} (by -\mathbf{r})
- equivalent statements of equilibrium
- \sum F_x = \sum F_y = \sum M_O = 0 (general coplanar)
- \sum F_x = \sum F_y = 0 (concurrent coplanar)
- concepts: forces, moments, couples, distributed loads, friction, tipping
- techniques: FBDs, internal reactions, method of joints, method of sections
