This paper focuses on model development for computer analysis of the thermal behavior of an externally driven spindle. The aim of the developed model is to enable efficient quantitative estimation of the thermal characteristics of the main spindle unit in an early stage of the development process. The presented work includes an experimental validation of the simulation model using a custom-built test rig. Specifically, the effects of the heat generated in the bearings and the heat flux from the bearing to the adjacent spindle system elements are investigated. Simulation and experimental results are compared and demonstrate good accordance. The proposed model is a useful, efficient and validated tool for quantitative simulation of thermal behavior of a main spindle system.
The accuracy of a machine tool is influenced by its static, dynamic and thermo-elastic behavior [
The spindle system’s overall behavior strongly depends on the design and structure of the employed components. Particularly, the behavior of high-speed spindles can vary significantly according to the direction or arrangement of its components. For example, the arrangement of bearing sets, motor placement and motor coolant jackets in the housing, fits between bearings, housing design and bearing preload mainly influence the spindle behavior during high speed operation.
Spindle bearing friction is the main reason for bearing heat generation, which limits the maximum achievable spindle speed. As speed increases, there is an increase in generated heat in the bearings, the motor and the cutting surface. This additional heat causes thermal expansion. The amount of heat generated in the bearing should be estimated within the design process in order to choose the proper types of bearings and drives.
Angular contact ball bearings are most widely used for high speed spindles due to their properties under high speed conditions [
Therefore, it is important to include the spindle’s and bearing’s thermal effects on the prediction of the overall response at elevated rotational speeds. The thermal models of bearing and spindle must be combined to provide a comprehensive representation of the heat transfer mechanisms.
Many attempts have been made to model the thermal behavior of the spindle and bearings for more than 60 years [
In order to achieve a better understanding of the thermal behavior of the main spindle and its influence on the surroundings, it is not sufficient to use specially designed test rigs for the testing of individual spindle components. Beyond that, an examination of the entire spindle system is required. For this reason a custom-designed modular test rig was built. Different machine tool spindles can be operated on this test rig under a variety of testing conditions. During the early stages of design and as a part of the comprehensive effort to define the error caused by thermal expansion due to spindle operation, a thermal model of the spindle system was developed, following the purpose of increasing the efficiency of evaluating the thermal behavior of such spindle systems. Using the model and software developed, a fast numerical evaluation of the effect of machining operation parameters on the heat transfer mechanism within the main spindle system was obtained.
For the purpose of model validation, an experimental study was conducted using a custom-built test rig. In this paper the test rig configuration and experimental work for the externally driven spindle’s thermal behavior are investigated. In this spindle as a special case study, the only heat generation sources are the bearings. The experimental work is followed by modelling and simulation of the thermal behavior of the spindle system using Wolfram Mathematica® software.
The purpose of the spindle bearing model is the calculation of heat generation, heat transfer and temperature distribution over the spindle and housing elements. The developed model is coupled with the thermal models of the bearing to obtain a thermal response of the whole spindle system. The following assumptions were made [
・ Shaft and housing are assumed to be radially axisymmetric about the centerline of the spindle.
・ The primary analysis will be one-dimensional in the axial direction. Radial heat loss in the housing will also be considered.
・ Any heat generation (or cooling) is assumed to occur at the bearing contact zone, the center of the spindle (as in a centrally located motor), or the tips of the spindle (cutting heat and motor heat).
Although there are many simplifications, this heat transfer model can be used to examine the temperature fields in the bearings, housing and spindle shaft by examining the actual spindle bearing assembly. Without loss of generality, the proposed heat transfer model is developed based on an externally driven grinding spindle with maximum spindle speed of 7000 rpm and a bearing bore diameter of 85 mm.
The externally driven grinding spindle investigated in this research is assembled with sealed universal bearings with small steel balls.
The heat is mainly generated in the contact between bearing raceways and balls due to frictional losses and rolling friction, influenced by speed, preload and lubricant. The cutting process is also a heat source. All three types of heat are the result of rotating motion and they are calculated from torque and speeds. The equation combining all three sources of heat is [
where
The bearing parameters
where D is the ball diameter (m),
where
It is assumed that for grease lubrication all heat generation enters the combined ball/ring network. The overall bearing heat generation is the summation of the contact heat generation at the inner and outer rings [
where
where k is the ball index,
The bearing displacement vector
where
As shown in
The following relations are derived from
The equilibrium equations for the bearing ball as shown in
where
For
The elements of the error vector
The bearing’s contact loads
The developed model is based on representing the spindle and the housing using axisymmetric thermal resistance elements. For the solution, the input parameters are: geometry, material parameters, air temperature, heat generation due to the bearings and initial temperature of the system. The thermal resistance is calculated for each housing and spindle element. Linear thermal resistance is defined according to the following equation [
where
The heat transfer equations use temperatures at the edge of the inner and outer rings as boundary conditions. As an alternative solution method, the outer ring temperature can be easily measured and applied to the solution, allowing the housing approximations to be ignored. Steady state outer ring temperatures can be used to provide a reference point for the thermal equations [
where
The resistance matrix elements,
The housing heat transfer model based on quasi-two-dimensional analysis of heat flow is shown in
where
ends in the element. In order to simplify the heat transfer equilibrium equation the axial and radial element resistance are combined as shown in the following equation:
where
As shown in
where
All matrix and vector elements presented in Equation (30) are expressed in Appendix D. By solving the set of Equation (30), the temperature values in six points of interest along the spindle shaft centre are obtained.
The configuration of the test rig used for experimental investigations of an externally driven spindle is shown in
The motor, friction in the bearings and the cutting process are the main causes of heat generated by the spindle system during machining operation. The motor is cooled using air flow in the direction from the drive end of the motor to the non-drive end. By assuming that most of the cutting heat is taken out by the coolant and that the heat generated by the motor in the configuration shown in
・ at the tool end: 1 spindle bearing set, in double O arrangement (Tandem), as fixed bearings
・ at the drive end: 1 spindle bearing set, in O arrangement, as floating bearings
The employed contact angle of 15˚ is suitable for high radial rigidity. The universal design bearings are lightly preloaded. The sealed spindle bearings require no maintenance and are lubricated for life with roller bearing grease [
Studying the heat flux mechanisms during spindle operation, the most interesting spindle elements are the spindle bearings, housing and shaft. In order to measure the temperature distribution in axial and radial direction and investigate the heat flux into the machine tool components adjacent to the spindle elements, the spindle system was divided into four cross-sections for temperature measurements.
The thermo-elastic behavior of a machine tool is strongly nonlinear and has to be modelled for several operating points. Hence, the speed spectrum has to be divided into several sections, each having its own operating point [
By changing the spindle speed over the speed spectrum, from 1000 rpm to 7000 rpm, it is clear from
several outer housings is investigated as well. By verifying the heat transfer from the inner bearings element to the outer housings, one is able to understand the heat flux mechanism and the influence of the heat generated in the bearings on the spindle-adjacent machine elements.
Based on the model described in Section 2 and Appendices A to D, a simulation program was developed. The numerical computations as well as the graphical presentations and investigations were carried out using the Mathematica® software [
Due to the fact that a commercial grinding machine spindle was used for the investigations, as described already in Section 3, in this stage of the experimental studies, we focused our experiments on the heat transfer mechanism from the bearings towards the outer housing elements as a function of spindle rotational speed. As stated, the spindle’s original geometry, preload, lubrication viscosity, structure or functional elements can not be varied during the investigations. The simulation results are compared with the results obtained in the experimental studies described in Section 4. The numerical simulations investigate the effect of operating conditions such as rotational speed (rpm), axial preload, lubricant viscosity, as well as geometrical parameters of the spindle system structural components. Comparison between measured temperatures at the housing outer surface, sensors 101, 201, 301 and 401, and simulated temperature along the axis of the spindle system in the bearings’ positions is shown in
From
Internal heat generation and heat flow is strongly speed-dependent.
The temperature profiles in
The theoretical and experimental investigations of the heat transfer mechanism in the externally driven grinding spindle have enabled the following conclusions.
1) The use of the bearing heat generation terms in each model, bearing model, spindle model and housing model enabled investigation of the thermal behavior of the entire spindle system under the effect of the main source of heat generation: the bearings’ friction.
2) Experimental investigation of the heat transfer in the spindle system and the machine elements adjacent to the spindle allows obtaining the data needed for better understanding the heat generation and flux due to the bearing friction.
3) The simplifying assumptions of the method presented in this study are mainly the one-dimensional heat flow approximations, simplified housing geometry, external drive motor, excluding the cutting process heat and excluding cutting process loads. These spindle-housing approximations are acceptable since the most important temperature predictions are those within the bearings.
4) Using the models presented, the temperature field can be numerically found for most spindle-housing configurations in a fast and efficient manner. The simulated and validated temperature fields can be used for numerical simulations of spindles thermal expansion and machine elements adjacent to the spindle, as well as for machine tool precision prediction during the development process.
The Authors want to thank the DFG (German Research Foundation) for financial support. The represented findings result from the subproject B03 “Investigation of Components and Assembly Groups” of the special research field SFB/Transregio 96 “Thermo-energetic design of machine tools”.
where
Housing free convection [