Abstract
This paper investigates dynamics of regenerative chatter in self-interrupted plunge grinding with delayed differential equations (DDEs) and partial differential equations (PDEs). The DDEBIFTOOL, a numerical simulation tool and the method of multiple scales are used to analyse stability and construct bifurcation diagrams. It was found out that in majority of cases, chatter is accompanied by a loss of contact. The loss of contact leaves uncut surface during the pass of grinding wheels, and thus the regeneration mechanism does not play a role. In that case, the delay used to represent the time span between two successive cuts should be multiple (double, triple, quadruple or higher). As a consequence, the chatter with losing contact cannot be accurately described by the DDEs with a fixed time delay. To address this problem, the PDEs are introduced to record the variation of workpiece profile. The PDEs are transformed into the ODEs by a Galerkin projection, and then the grinding dynamics is studied numerically. Solutions obtained from the DDEs and the PDEs are in a good agreement for a continuous grinding but there is a discrepancy for a self-interrupted cutting.
Highlights
Chatter-free operation is critical for grinding as it guarantees a good workpiece surface finish and prolongs life of a cutting tool
The regenerative machining dynamics has been investigated by using the regenerative chatter theory, which is mathematically described by delayed differential equations (DDEs) [6, 7, 15, 17, 24]
A further comparison between the results obtained from the DDEs and the partial differential equations (PDEs) is given in bifurcation diagrams depicted in Fig. 11, which were constructed from numerical simulations of the stable grinding processes
Summary
Chatter-free operation is critical for grinding as it guarantees a good workpiece surface finish and prolongs life of a cutting tool. To describe the regenerative effect during grinding, the DDEs are employed, where the cutting force is modelled by the instantaneous grinding depth or in another word the chip thickness [2, 39]. A delay term represents the previous relative displacement between the tool and the workpiece in the previous turn, the DDEs can be naturally employed to model the regenerative dynamics. This idea was first introduced by Tobias [34], who investigated the regenerative turning chatter. In comparison with the supercritical chatter, the subcritical one induces vibration with much larger amplitude by generating a negative grinding depth, which means a lost of contact between the grinding wheel and the workpiece. When the wheel loses contact with the workpiece, a discrepancy increases with time
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