Chapter 4, Problem 70P

Figure P4. I 7 shows the free-body diagrams for planetary gear components used in the variable valve timing (WT) system of an internal combustion engine (Ren, 2011). Here an electric motor is used to drive the carrier. Analysis showed that the electric motor with planetary gear load (Figure P4.l7) may be represented by the following equation:

${\Omega }_c\left(s\right)=G_e\left(s\right)E_a\left(s\right)+G_m\left(s\right)T_{cam}\left(s\right)$

where ${\Omega }_c\left(s\right)$ is the output carrier angular speed in rad/s, $E_a\left(s\right)$ is the input voltage applied to the armature, and $T_{cam}\left(s\right)$ is the input load torque. The voltage input transfer function, $G_e\left(s\right)$ , is

$G_e\left(s\right)\cong \frac{K_{\tau }}{R_m\left(Js+D\right)+K_{\tau }{K}_m}=\frac{45}{0.2s+1}$

and the load torque input transfer function, $G_m\left(s\right)$ , is

$G_m\left(s\right)\cong \frac{-R_{m}k}{R_m\left(Js+D\right)+K_{\tau }{K}_m}=\frac{-5}{0.2s+1}$

Find an analytical expression for the output carrier angular speed, ${\omega }_c\left(t\right)$ , if a step voltage of 100 volts is applied at t = 0 followed by an equivalent load torque of 10 N-m, applied at t = 0.4 sec.

FIGURE P4.17 Free-body diagrams of planetary gear system components¹²