(untitled) (Innovation in Woodworking Industry and Engineering design, 2026)
The proposed study deals with the modeling of the free spatial vibrations of a woodworking shaper and its spindle. An original mechanical - mathematical model of woodworking shaper and its spindle, developed by the authors, is presented in this work. The model provides the opportunity to explore the free undamped space vibrations of this type of machinery. In this model the woodworking shaper and its spindle are regarded as rigid bodies, which are connected by elastic elements with each other and with the motionless floor. A system of matrix differential equations is compiled and analytical solutions are presented. The natural frequencies and mode shapes for a specific machine can be obtained with their help. The model is supplemented in order to investigate the free damped space vibrations of the considerate system. The new model takes into account the damping properties of machine’s elements. A new system of matrix differential equations is developed and the relevant analytical solutions are presented. They allow various numerical solutions for specific machines. The models take into account the characteristics in the construction of woodworking shapers. They include all necessary geometric parameters of this system. Numerical solutions and graphs can be obtained with the help of the developed models. These solutions and graphs are necessary for analyzing the free undamped and the free damped space vibrations of a particular considerate machine.