D'Alembert's principle; gears with couple

D'Alembert's principle converts a dynamics problem into a statics problem. A body in motion is in dynamic equilibrium if, in addition to the real forces and moments, we add inertia effects equal and opposite to the effective forces:

• Inertia force $-m\bar{\vec a}$ acting at the mass centre $G$.
• Inertia couple $-\bar I\vec\alpha$ (for plane motion).

For a rigid body in plane motion the equations of dynamic equilibrium are

$$\sum F_x - m\bar a_x = 0, \qquad \sum F_y - m\bar a_y = 0, \qquad \sum M_G - \bar I\alpha = 0$$

which are the same as $\sum \vec F = m\bar{\vec a}$ and $\sum M_G = \bar I\alpha$, but written as "sum of forces including inertia terms is zero". The moment equation can then be taken about any point, as in statics. Special cases: pure translation ($\alpha=0$): only the inertia force; rotation about a fixed axis through $G$: only the inertia couple $-I\alpha$.

   real forces + (-m a) at G + (-I alpha) couple  =>  equilibrium

Numerical part

Data: $m_A = 10$ kg, $\bar k_A = 0.080$ m, $r_A = 0.250$ m, $r_B = 0.100$ m, $M = 8$ N m on gear $B$. The data give no mass for gear $B$, so its inertia is neglected (if it is included, add $I_B$ in the equations below). The tangential force between the teeth is $F$.

$$\bar I_A = m_A\bar k_A^2 = 10(0.080)^2 = 0.064\ \text{kg m}^2$$

        M (on B)
         ( B  )---F--->(tooth)
                      ( A )

(b) Tangential force on A

Gear $B$ (massless): $M - F r_B = 0$

$$F = \frac{M}{r_B} = \frac{8}{0.100} = 80.00\ \text{N}$$

(a) Time to reach 600 rpm

Gear $A$: $F r_A = \bar I_A\alpha_A$

$$\alpha_A = \frac{F r_A}{\bar I_A} = \frac{80.00\times0.250}{0.064} = 312.50\ \text{rad/s}^2$$

The teeth have the same tangential acceleration: $\alpha_B r_B = \alpha_A r_A$

$$\alpha_B = \alpha_A\frac{r_A}{r_B} = 312.50\times\frac{0.250}{0.100} = 781.25\ \text{rad/s}^2$$ $$\omega_B = 600\ \text{rpm} = \frac{600\times2\pi}{60} = 62.83\ \text{rad/s}, \qquad t = \frac{\omega_B}{\alpha_B} = \frac{62.83}{781.25} = 0.0804\ \text{s}$$

Answer: $t = 0.0804$ s; tangential force $F = 80.00$ N.