Problem $4$: This is a problem about gear design and profile$-$shifted gears. Solve each problem.

$(1)$ Pinion: Material SM$45$C $($HB$250),$ rotational speed $1000$ rpm

Gear: Material GC$150,$ rotational speed $500$ rpm

The center distance is approximately $600$ mm$,$ the pressure angle is $20,$ and the face width is $5$m $($module $=$ m$).$ Calculate the physical size of the teeth when the pair of standard spur gears transmits $10$ kW of power with shocks. $($Note: Use the textbook's table for the speed factor and shock factor.$)$

$(2)$ Given a module m$=4$mmm $=4\backslash ,\backslash$text$\{$mm$\}$m$=4$mm$,$ number of teeth Z$1=20$Z$\_1=20$Z$1=20,$ Z$2=40$Z$\_2=40$Z$2=40,$ and pressure angle of $20,$ and profile$-$shifted gears with profile shifts x$1=0\mathrm{.}5$x$\_1=0\mathrm{.}5$x$1=0\mathrm{.}5,$ x$2=0\mathrm{.}4$x$\_2=0\mathrm{.}4$x$2=0\mathrm{.}4$: $(1)$ Calculate the center distance increase factor and the center distance of the profile$-$shifted gears.

$(2)$ Calculate the root circle diameter and addendum circle diameter of the profile$-$shifted gears. $($Gap factor k$=0\mathrm{.}157$k $=0\mathrm{.}157$k$=0\mathrm{.}157).$