Kinetic Analysis

D’Alembert’s principle

By this principle, a particle or a system of particles moving with a constant acceleration is known as in dynamic equilibrium.

For a particle

$$F - ma = 0$$

Here:

• $F$ - resultant applied force
• $m$ - mass of the particle
• $a$ - acceleration of the particle

$ma$ is known as the inertia force which is fictious. $$

For a system of particles

The equation can be rewritten as:

$$F = \sum m_ia_i = F = ma_G$$

Here:

• $F$ - resultant applied force
• $m_i$ - mass of a particle
• $a_i$ - acceleration of a particle
• $m$ - total mass
• $a_G$ - acceleration of the center of mass

Kinetics of a Rigid Body

Both translational and rotational aspects of the motion must be considered.

$$F=ma \;\;\; \text{and} \;\;\; M=I\alpha$$

Here:

• $F$ - resultant applied force
• $m$ - mass of the body
• $a$ - acceleration of the body
• $M$ - moment of the resultant force
• $I$ - mass moment of inertia
• $\alpha$ - angular acceleration of the body

General plane motion

$$\sum M_G = I_G \alpha$$

Here:

• $M_G$ - moment about center of mass
• $I_G$ - mass moment of inertia about center of mass
• $\alpha$ - angular acceleration of the body

Let $O$ be a point other than $G$.

$$\sum M_O = I_G \alpha + ma_G r_{OG}$$

Kinetic energy

$$T = \text{translational kinetic energy} + \text{rotational kinetic energy}$$ $$T = \frac{1}{2}mv_G^2 + \frac{1}{2}I_G\omega^2$$

Center of percussion

The point on an object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot.

Translational and rotational motions cancel at the pivot when an impulsive blow is struck at the center of percussion.

$$l_{\text{P/G}} = \frac{k_\text{G}^2}{\text{OG}}$$

Here:

• $\text{P}$ - Center of percussion
• $\text{G}$ - Center of gravity
• $\text{O}$ - Pivot

For a uniform rod:

$$l_{\text{P/G}} = \frac{L}{6}$$