Research On Coupled Vibration Characteristics Of The Spindle Unit In The Electric Spindle With Spectral-geometry Method

The electric spindle is a core functional component of CNC machining equipment.The coupled vibration of the spindle unit is one of the important factors affecting the machining accuracy of the electric spindle.It is important to establish the transverse,longitudinal,and torsional coupled vibration models of the spindle unit for the quantitative analysis of machining accuracy.In the study of vibration characteristics,the spindle unit is mostly simplified as an elastic beam model.In order to realize the parametric research and boundary stiffness sensitivity research of the elastic beam model.In this paper,the following research work is carried out on the dynamic modeling problem of the elastic beam model:First,the spindle unit in the electric spindle is simplified as an elastic single-segment beam.A coupled model is established to analyze the transverse,longitudinal,and torsional vibration of elastic single-segment beams under arbitrary boundary conditions based on the Spectro-Geometric method.The vibration displacement functions of single-segment beams are respectively expressed as improved Fourier series.The boundary restraining springs are introduced at both ends of the single-segment beam and the stiffness value of springs are changed to simulate arbitrary boundary conditions.Hamilton’s principle is employed to derive the Lagrangian function of the structure and the Ritz method is used to get the solution.The first 7 order natural frequencies under different boundaries of beams are calculated.When the series truncation number N takes 12,with the maximum error of 1.51% compared with literature solutions and finite element solutions.The correctness and fast convergence of the method are validated.The displacement function representation form and modal characteristic solution equations of the transverse,longitudinal,and torsional vibration of single-segment beams are unified.Then,the spindle unit in the electric spindle is simplified as an elastic multi-segment beam.A coupled model is established to analyze the transverse, longitudinal,and torsional vibration of elastic multi-segment beams under arbitrary boundary and coupling conditions based on the Spectro-Geometric method.The vibration displacement functions of multi-segment beams are respectively expressed as improved Fourier series.The restraining springs are introduced at both ends and couplings of multi-segment beams and the stiffness value of springs are changed to simulate arbitrary boundary and coupling conditions.Hamilton’s principle is employed to establish the Lagrangian function of the structure and the Ritz method is used to get the solution.The first 7 order natural frequencies under different boundaries of multi-segment beams are calculated.When the series truncation number N takes 12,with the maximum error of 0.27%compared with literature solutions and finite element solutions.The correctness and fast convergence of the method are validated.The displacement function representation form and modal characteristic solution equations of the transverse,longitudinal,and torsional vibration of multi-segment beams are unified.Based on the modeling of transverse,longitudinal and torsional vibration of the elastic multi-segment beam model,the parametric research and boundary stiffness sensitivity research of the elastic multi-segment beam model are carried out.Using the established elastic multi-segment beam coupled vibration analysis model,the influences of the multi-segment beam geometric dimensions,material parameters,and the stiffness coefficients of the restraining springs at both ends and coupling locations on the transverse,longitudinal and torsional vibration characteristics of the multi-segment beam are discussed.The researched influence laws can provide a reference for the dynamic design of the structure and the realization of the vibration reduction target,and provide a parametric research method for the optimization of the dynamic performance of elastic multi-section beams.