Mathematical Model of the Tensioning in the Collet Clamping Mechanism with the Rotary Movable Input Link on Spindle Units

Abstract

Increasing machining productivity causes the cutting forces acting on tools or workpieces to grow and requires extra clamping forces for their fixation reliably. In the research, a mathematical model of the operation of the clamping mechanism for fixating cylindrical objects on the spindle of machine tools at the stage of tension is presented. The presented design of the mechanism contains screw gear and provides self-braking. Based on the calculation model, mathematical dependencies are developed to describe the relationship among the movements of the parts of the mechanism when clamping forces are growing. The presented analytical dependencies allow considering the stage of growing clamping forces separately when the conservative type of forces are prevailing in the mechanism’s operation. That stage of work when both types of forces of dissipative and potential characters exist is considered. The developed dependencies describe the position of parts of the clamping mechanism depending on the generalized coordinate. The angle of rotation of the input rotating link is used as the generalized coordinate. This fact allows calculating the position of the elements of the clamping mechanism of this type depending on time. Results of the research enhance understanding the pattern of the change in the interaction of the elements and forces that act in the mechanism during the final stage of clamping. The obtained mathematical dependencies are a precondition for the development of design methodology for mechanisms of this type.