The work deals with consideration of the technology of forming many-sided outer surfaces with the variable profile of planetary turning. The peculiarities of this technology were detected and the actuality of the research problem was specified. The essence of study lies in the determination of the cutting force’s effect on the blank rigidity. The calculation scheme and mathematical model, which allow studying the cutting forces effect on the rigidity of technological system during processing, were stated. The dependence of blank rigidity on the form of cross section processing with multiple-blade instrument with planetary movement was determined. The mathematical expression of the allowable twist angle under effect of cutting forces on the blank was offered. This study can be applied as the test calculation at projecting of technological rigging and setting modes of cutting at forming of many-sided outer surfaces with variable profile of planetary turning.