A simple lathe structure can be represented as a single$-$degree$-$offreedom system aligned along the direction of the normal to the cut surface, with an equivalent mass $m_e,$ damping coefficient $c_d,$ and stiffness $S_e.$ If the direction of the cutting force is inclined at an angle ${}_c,$ to the normal to the cut surface, show that the limit of stability for fully regenerative chatter is defined by the following equations:

$\frac{k_1}{S_c}=\frac{-(1-(\frac{}{{}_n}{)}^2)}{(1-cos{}_c)cos{}_c}$

Machine Tool Vibrations

$317$

FIGURE $9\mathrm{.}25$ Width of cut at stability boundary for various differences in helix angles on helical milling cutters $($rotational speed $=122rpm).($From Opitz, H$.,$ Dregger, E$.$U$.,$ and Roese, H$.,$ Proc. $7$th Int. MTDR Conf., Pergamon Press, New York, $1966.)$

and

$\frac{k_1}{S_f}=\frac{-2{}_{}^{}}{{}_{n}sin{}_{c}cos{}_c}$

where ${}_c=T,T=$ the period between successive passes of the tool, ${}_n$ is the natural frequency of the system, and $$ is the damping ratio. It should be assumed that the cutting force varies only due to variations in the undeformed chip thickness and $k_1$ is the chip thickness coefficient $($or cutting stiffness$).$ If ${}_n=50Hz,=0\mathrm{.}5,$ and ${}_c=60$ degrees, plot the highest speed unstable region of the stability chart in terms of $\frac{k_1}{S_e}$ and ${}_c.$