Abstract
A machinetool structure optimization is an important technique that improves the machining efficiency and saves materials and the energy resource. In this work, dynamic design optimization method for MachineMotorizedSpindle (MMS) subjected to a number of rotating unbalanced forces effects is presented. Linear forcedRotordynamic analysis with design explorer method has been used to simulate the output response. The Design Variables (DVs) and their limits were carefully chosen and applied to develop the DesignofExperiment (DOE). The BoxBehnken Design (BBD) method, because of its good organization in providing much information in a minor number of required statistical experiments was used to generate the DOE. The influences of DVs on the dynamic of MMS and their levels optimization were evaluated by utilizing the ResponseSurface (RS) method. The results showed that the spindle shaft inner diameter of the motorrotor seat and its rotating unbalanced mass, and modulus of elasticity have the highest contribution in effect on the dynamic of MMHS. As well, it is found that the proposed optimization method not only improves the structural weight of MMS, but also the potential saving can be achieved in term material and energy resource.
Highlights
 Parametric Optimization problem
 Design of Experiment for design space exploration
 Response Surface model
 Check the quality of Response Surface model, Sensitivities analysis
 Optimization based on Response Surface model
1. Introduction
The machinetool spindle has an important role in machining process, since it provides a relative movement among the tool and workpieces, and also, the torque required to remove material. In general, the rotating unbalance forces induced due to workpiecesholder and motorrotor will subsequently cause the vibration of the spindle. Thus, the vibration behaviors of unbalanceresponse remain a foundation of a research for vibration observing and balancingcontrol [1]. Because of complex structures of MMS, the evaluation of dynamiccharacteristics produced by rotating unbalance force is much difficult [2]. The vibration response caused by rotatingunbalance force induced on the spindlesystem was widely investigated by considerable researches [3, 4]. Zhou et al. [5], conducted a study on motorrotor unbalance effects for identifying damping and stiffness coefficients of closedloop activemagneticbearing. Wu et al. [6], comprehensively introduced the influences of motorrotor eccentricity on surface topography in ultraprecision machine, by using a new integrated electromechanical coupler manner for estimating unbalanced force in airmotorizedspindle. In MMS design, there are many factors influence its dynamic performance; so, it important to define the levels of these factors in design stage. A machinetool structure shape and dimensions optimization is an important technique for saving an energy and material resource, as well as improving the machining efficiency. Considerable researches have been conducted to improve the dynamiccharacteristics of MMS [713], through many optimization categories such as spindleshaft and motorrotor specifications; bearings configuration and number of bearings; bearings preload. Most of these previous works are focused on enhancing the dynamic performance of spindlesystem under unforced vibration induced. However, the work that addresses the linear forced rotordynamic analysis with design optimization for MMS subjected to unbalance forces effects is comparatively fewer. In the machining operation, the HighSpeed (HS) effects and rotating unbalance forces are introduced, hence, it should be coupled into FEM to precisely predict the dynamic performance of MMS. The Rotordynamic analysis is the most effective methods in introducing the HS effect, crosscoupled loads, and possibility of rotating instability. In this work, dynamic design optimization method for MMS subjected to a number of rotating unbalanced forces effects is considered. Linear forcedRotordynamic analysis with design explorer method is adopted to simulate the outputresponse. The inner diameter and its material properties of MMS, and rotating unbalanced mass of motorrotor and chuck are considered as the main DVs for optimization problem. The DVs limits were carefully chosen and utilized to develop the DOE for designspace exploration. The BBD method, because of its best efficiency in providing much data in a minor number of required statistical experiments was used to generate the DOE. Then, the effect of DVs on dynamic performance of MMS and their levels optimization were evaluated by employing the RS method. Finally, the efficiency of RS analysis is tested by comparing the FEM results from the present work with the experimental results, which carried out by Huang P., et al [3]. In present work, the Cartridge Spindle for CN lathe, L series standard JIS A26 that developed in literature [4] is considered to establish the FEM model.
1.1. Forced rotordynamic analysis
Rotordynamic is the vibration of rotating machinery that conducted at the present of gyroscopic moment’s effects, and unbalanced forces. The governed system equation of motion in generalized matrix form, for an axially symmetric rotor rotating at spin speed $\mathrm{\Omega}$ is given as follows:
where $M$: symmetric mass matrix, $D$: symmetric damping matrix, $G$: the skewsymmetric gyroscopic matrix, ${K}_{B}$: the symmetric bearing stiffness matrix, ${K}_{b}$: the symmetric beam stiffness matrix, $N$: the gyroscopic matrix of the deflection due to centrifugal force. The Eq. (1) numerically can be computed by any of a massive amount of existing linearequation solvers [4]. In this study, direct integration method was used to find out the steadystate FrequencyResponseFunction (FRF) through using ANSYS software.
2. The description of finite element model
To predict the dynamic response with all sensational loads, the mathematical models of chuck (chuck standard B6151sc) and motorrotor were coupled with the FEM model. For simplification, the models of motorrotor and chuck were displayed as a point mass (MASS21element) with remote type connection under inertia load effects. The bearings were modeled as 2D elastic springdamper element (COMBI214) with groundtobody Type connection. The locations of point mass and bearings on 3D model of MMS for FEM analysis is shown in Fig. 1. The bearings stiffness was estimated according to the bearings specifications, which is 4.5E5 N/mm for front bearing and 3.5E5 N/mm for back bearing. To fix degree of freedom in axial direction; the displacement constraint was added at bearings positions. The entire FEM has been meshed by considering the magnitude and accuracy of the design, which created the mesh size of 10 mm for whole FEM with type element size. The analysis was carried out under excitations of rotating mass of chuck and motorrotor within the excitation frequency range of 1004000 Hz.
Fig. 1The location of point mass and springdamper element on 3D FEM model
2.1. Parametric optimization
In the final design stage, a design’s performance is greatly inﬂuenced by its shape and size. Parametric optimization (PO) can be used to help designers for determining the optimal shape and dimensions of a structure. In general, PO involves minimizing an objective function of the DesignVariables (DVs) subjected to a given set of design constraints. So, the FEM developed in Section 2 was used to generate the DOE. The experiments were conducted according to the constraints displayed in Eq. (2) to determine structure Weight ($w$) and MaxDeformation (MD). The definitions of the DVs used for these experiments are reported in Table 1. The experiments were conducted through 65 experimental runs based on BBD experimental method. After repeating the experiments many times, the best results have been considered for RS evaluation:
To find the levels of $DVs=\left[{x}_{1},\mathrm{}{x}_{2},\mathrm{}{x}_{3},{x}_{4},\mathrm{}{x}_{5},{x}_{6},\mathrm{}{x}_{7}\right]$.
That minimize the weight ($M$) $M=\sum _{k=1}^{N}{\rho}_{k}{{\rm A}}_{k}L$, where: $\rho $ is density, $A$ is area, $L$ is length, and $N$ is the element number.
The subjected constraints are: $MD\le $ 0.032, $\pm 0.25{x}_{1}$, $\pm 0.25{x}_{2}$, $\pm 0.18{x}_{3}$, $\pm 0.05{x}_{4}$, $\pm 0.054{x}_{5}$, $\pm 0.2{x}_{6}$, $\pm 0.25{x}_{7}$.
Table 1The DVs and their definitions
DVs  The definition 
${x}_{1}$ [44 mm]  The motorrotor seat spindle shaft inner diameter 
${x}_{2}$ [44 mm]  The rear bearings seat spindle shaft inner diameter 
${x}_{3}$ [51 mm]  The front bearing seat spindle shaft inner diameter 
${x}_{4}$[200 GPa]  The young’s modulus of elasticity 
${x}_{5}$[7450 kg·m^{3}]  The density of the structural material 
${x}_{6}$ [12 kg]  The rotating unbalanced mass of the workpiecesholder 
${x}_{7}$ [15 kg]  The rotating unbalanced mass of the motorrotor 
3. Results and discussions
3.1. Effects of DVs on forced vibration response
In this study, the RS method is used to evaluate the effect of DVs on the forced vibration response. The goodness fit curve shown in Fig. 2 is plotted to test the quality of RS evaluation for $W$, and MD. Fig. 2(a) showed that the most of points are on the underline, as well in Fig. 2(b) all the verification points are greatly close to the observedvalues. Therefore, the RS model is able to predict most values of design points including the verification points. The contributions of the variables on the output response are plotted in Fig. 3, which shows the DV with higher input percentage has a robust effect on the $W$ (P16), and MD (P17). Fig. 3(a) showed that the effect of factor ${x}_{1}$ on the output response is the first, the factor ${x}_{6}$ is the second, the factor ${x}_{3}$ is the third, and factor ${x}_{2}$ is the fourth. Similarly, in Fig. 3(b), it can be noted that the outcome of factor ${x}_{1}$ on the MD is the first, the factor ${x}_{7}$ is the second, and the factor ${x}_{4}$ is the third. Among these factors the contribution of ${x}_{1}$ on output response is the largest when compared to others DVs. Accordingly, the motorrotor seat inner diameter (${x}_{1}$) and material density (${x}_{5}$), and rotating unbalancemass (${x}_{7}$) are the vital factors that entirely control the output responses for MMHS.
Fig. 2The predicted from RS vs. observed from the design points: a) learning points, b) verification points
a)
b)
Fig. 3The effects of DVs on output response: a) the structure weight, b) the maxdeformation
a)
b)
3.2. The optimum FEM
The optimization draws information from the RS; which dependent on the quality of RS evaluation. There are many RSbased optimization methods, such as Screening and MaltiObjectiveGeneticAlgorithm (MOGA). An iterative MOGA provides more refined method than Screening; therefore, it’s more suitable for calculating global min\max. By considering MOGA method, the objective function and constraints were adjusted based on Eqs. (7, 8). The algorithm has generated the three candidatespoints. The optimal output responses and their variation from the reference were reported in Table 2. From Table 2, it can be seen that there are percentage variation for output responses with regard to initial reference in candidatespoint 1, and 2, since its zero in candidatespoints 3. The vibrationbehavior of optimum FEMs were obtained by creating and updating (real calculation) the DVs of candidatespoints with verification point as shown in Fig. 4. The outcome obtained regard to the real solve for verification shown in Fig. 4(d) is comparable to RS optimization results of Fig. 4(a), (b), and (c). In Fig. 9, also it showed that the maximum vibrationresponse calculated is at motorrotor rotating force position, while it minimum at front bearings position. Based on the smallest percentage variant detailed in Table 2, the candidatespoint 3 remains the best solution that satisfied the design requirement with least computational error. To identify the efficiency of RS model, the optimal FEM is compared with the reference point as plotted in Fig. 5. From Fig. 5, it can be noted that the damped FRF is comparable to its initial model. The FRF improved in span between bearings, while it increased in spindle nose. As results, the structuralweight regard to the spindle shaft has been decreased by 12.5 % when compared to its reference weight (18.3 kg). Thus, the proposed optimization method is significant in terms of materials and energy resources saving.
Table 2The output responses and their variation from the reference
Optimal Points  Structure Weight  Maxdeformation  
Response  Variant  Response  Variant  
1  16.033 kg  –0.14 %  0.0289 mm  0.22 % 
2  16.043 kg  –0.07 %  0.0286 mm  –0.62 % 
3  16.055 kg  0.00 %  0.0288 mm  0.00 % 
Verification  16.066 kg  0.05 %  0.0274 mm  –0.94 % 
Fig. 4The FEMs mode shapes after optimization
a) First candidatespoint
b) Second candidatespoint
c) Third candidatespoint
d) DObased verification point
Fig. 5Compares initial and optimal behavior of FRF, a) span between bearings, b) spindle nose
The machine tool spindle design requirement for stiffness can be used to verify the FEM, by comparing the values of dimensionless amplitude of span between bearings. So, the confirmation of FEM is done by comparing numerical solutions to related results carried out in previous work [3]. First, the Static Deformation (SD) of span (${L}_{s}$) between bearings is calculated using the theoretically constrained function of $SD\le $ 0.0002${L}_{s}$. Secondly, the MaxDeformation (MD) of span between bearings is defined for finite element model at lower operating frequency. Then, the dimensionless amplitude is determined by computing the ratio of MD to SD. The computational results indicated that the FEM developed is comparable to experimentally validated results found in [3]. Wherein the dimensionless amplitude calculated are 0.34 for presented FEM, and 0.76 for publication [3], the slightest variation noted in finding may turned into a different in boundary conditions.
4. Conclusions
In this work, dynamic design optimization method for MMS subjected to a number of rotating unbalanced forces effects is considered. Linear forcedRotordynamic analysis with design explorer method is adopted to simulate the outputresponse. The DVs and theirs constrains were definitely selected and applied to develop the DOE for designspace exploration. The BBD method, because of its good organization in providing much information in a minor number of required statistical experiments was used to generate the DOE. The effect of DVs on dynamic performance of MMS and their levels optimization were evaluated by utilizing the RS method. The results showed that the spindle shaft inner diameter of the motorrotor seat and its rotating unbalanced mass, and modulus of elasticity have the highest contribution in effect on the dynamic of MMHS. The structuralweight of spindle shaft has been improved by 12.5 % saving when comparing to its initial weight. As well, it is found that the proposed optimization method not only improves the structural weight of MMHS, but also the potential saving can be achieved in term material and energy resource. Finally, the FEM has been confirmed through related experimentally validated results in the literature.
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About this article
The authors gratefully thank the technical support that provided by College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin, China, during this work.