A simplified linear dynamic grinding system can be represented by masses, springs, and elements. Based on the following simple grinding force model, which says that the normal grinding force Fn is proportional to the material removal rate damping
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where X is constant, b is grinding width, vw is workpiece speed, vs is grinding speed, a is wheel depth of cut, є is exponent, and x is mutual approach speed between grinding wheel and workpiece. The plus sign in the denominator is for external cylindrical grinding, the minus sign for internal grinding, and dw = for surface grinding.
kg, given by Equation 8.5, is the grinding stiffness, which is the coefficient between the grinding force and the depth of cut. cg, given by Equation 8.6, is the grinding damping, which is the coefficient between the grinding force and the mutual approach speed between the grinding wheel and the workpiece. The grinding damping increases in proportion to the length of the contact between the wheel and the workpiece. Generally speaking, the grinding system becomes less stable as the grinding stiffness increases, while increase of grinding damping makes the system more stable.
In order to analyze chatter vibration caused by the grinding wheel regenerative effect, the grinding stiffness given by Equation 8.5 should be replaced with the wear stiffness of the grinding wheel. As a first-order approximation, the wear stiffness is obtained as
k = k G—
s g v
w
where G is the grinding ratio. Taking practical values of the grinding ratio and the speed ratio into account, it is confirmed that the wear stiffness of the grinding wheel is much higher than the grinding stiffness. Therefore, as far as the analysis of the chatter vibration caused by the workpiece regenerative effect is concerned, the wear stiffness of the grinding wheel can be assumed to be infinite.