The vibration of machine motorized spindle based on forced rotordynamic analysis and response-surface method

2.1. Finite element model of machine motorized spindle system

The MMS exposing several loads under operation condition is established, including the rotating unbalance moment to the shaft from the motor rotor and constant cutting force. The FEM of the MMS is developed by using the ANSYS SpaceClaim. All the masses of motor-rotor and workpieces-holder (chuck) are modeled as a point mass and integrated into FEM at desired locations, with a Remote-Point-type connection. The bearing supports are modeled as 2-D spring damper elastic elements (COMBI214) with a ground-to-body-type bearing connection, as shown in Fig. 2.

Fig. 2Bearings support

Fig. 3The FEM meshing for MMSshaft

Two spring-damper elements for supporting the shaft at the front bearing set and so one spring-dampers element for supporting the shaft at back bearing set. With assuming that, the applied preloads on bearings are not influenced by their locations; the bearing stiffness defined in this study is 4E5 N/mm for the front bearing set and 2E5 N/mm for back bearings set. The entire spindle model has meshed via elements size of 10 mm, the 3D meshing of MMS is shown in Fig. 3.

3.1. The unbalance response analysis

The forced Rotordynamic analysis is performed under two excitation conditions, first under only constant cutting force and second under both cutting and rotating unbalance forces excitation. The simulation is conducted with the 3700 N excitation cutting force applied in the $y$-direction, and excitation rotating unbalance of 6 kg-mm, with excitation speed varied from 0 Hz to 400 Hz. The simulations results undercutting force, and with both cutting force and rotating unbalance excitations are shown in Fig. 4. It can be seen from Fig. 4(a), the maximum Total Deformation (TD) is 0.0496 mm, which it recorded at spindle nose; while the maximum TD in Fig. 4(b) is 0.0509 mm; which it takes place in the motor-rotor location. The results obtained for TD indicated that the spindle has extremely affected more by the rotating unbalance force excitation than the cutting force excitation.

In order to visualize the effect difference excitation speed on unbalancing response, the four locations of the MMS as shown in Fig. 5 have been selected for vibration analyzes. From Fig. 5, it can be seen that; the vibration amplitude has been caused by the rotating unbalance in the MMS system. Among the four places, the vibration response at motor-rotor seat position is far greater than other places in the spindle system. Once the deformation amplitude of the MMS is growth, the dynamic stiffness is turns into lower. Thus, the minimum dynamic stiffness in Fig. 5 is found at motor rotor place, while the maximum dynamic stiffness is found at spindle nose position. As well, the vibration modes increased in a linear manner as the excitation frequency increase, which is relatively comparable to theoretical studies [3]. The vibration response reached its peak at 400 Hz excitation frequency. There is no vibration resonance recorded within the excitation frequency range, as result of damping effects. As mentioned above, the vibration of MMS is considerably influenced by the rotating unbalance due to motor-rotor, hence it’s significant to consider in FEM when conducting the vibration design.

Fig. 4Total deformation of the spindle: a) undercutting force excitation, b) under rotating unbalance mass and cutting force excitation

a)

b)

Fig. 5The vibration modes shape of unbalance response at different locations on the spindle: a) mode shape at rotor position, b) mode shape at spindle nose, c) mode shape at the front bearing location, d) mode shape at back bearing location

a)

b)

c)

d)

3.2. Sensitivities of factors on vibration of unbalance response

In this study, the FEM-based experiment has been carried out to simulate the vibration response and SW with the factors. The definition of the factors used in this experiment and their limits are shown in Table 1. The face-centered CCD due to its effectiveness in providing much statistics in a minimal number of required statistical experiments is used to create the DOE. The simulation result obtained is fitted by RS, and then it graphically explained in Fig. 6. It showed that the scatter chart presented is normalized with the output values predicted from the RS versus the value observed from the DOE. The goodness fit indicated that the TD and SW are normally closer; confirming that the RS is correctly fits the points of DOE.

Fig. 6The predicted response surface output versus observed from DOE

3.2.1. The levels of factors affecting the unbalance response

The sensitivities of the factors on TD and SW are analyzed by using Kriging RS method. The contribution of each factor on TD and SW is graphically represented in Fig. 7. From Fig. 7, it can be showed that a larger response ratio indicates that the corresponding factor has a more obvious influence on the output responses of TD (P1), and SW (P25). For the station P1shown in Fig. 7, the factors $Z_{10}$ (45 %), $Z_2$ (20 %), $Z_4$ (18 %), and $Z_6$ (11 %) have the most effective on TD, respectively; while the other factors have the least effects on the TD. For the station P25 shown in Fig. 7, the factors $Z_2$ (50 %), $Z_7$ (25 %), $Z_3$ (15 %), and $Z_1$ (12 %) have the most effective on SW, respectively; while the other factors have the least effects on the SW. As results, more than 40 % of vibration is conducted by rotating unbalance mass ($Z_{10}$), hence the rotating force excitation is far greater than cutting force excitation ($Z_9$). In order to visualize the effect of variation of rotating unbalance mass on the dynamic response, the change in TD with the factor $Z_{10}$ is plotted as explained in Fig. 8. It can be seen from Fig. 8, the variation of TD with the factor $Z_{10}$ is approximately nonlinear, which the TD improved/and or increased with increasing the factor space along its variation limits. The variation of vibration response with rotating unbalanced masses at different rotating speed has carried out in [3], they obtained that the dynamic response of the MMS is increased linearly with increasing the rotational speed, and so as rotating force increased.

Fig. 7The sensitivities of factors on the vibration of unbalance response

Fig. 8The effect change of rotating unbalance force on TD

3.2.2. Factors optimization

The definition of the optimization problem and its constraints are represented in Eq. (1). The optimization is defined to minimize the SW, with a maximum constraint of TD (0.067 mm). The Screening and Multiple-Objective Genetic Algorithm (MOGA) methods are used to solving this mathematical problem. The three optimal’ points for each method are suggested, and then they are summarized in Table 2. It can be observed that the optimal points in columns 3&7 of Table 2 are feasible solution that could meet the design requirements. Among them, based on cutting force criteria the design points in column 7 is most suitable in minimizing the SW with satisfying the dynamic stiffness requirements. Further, from the response points shown in Table 2 it found that there are significant improvements achieved for the TD and SW, when comparing the optimal design with the original value. It showed that the SW is saved by 16.8 %, while the TD is improved by 13.7 %. So as to find the dynamic behavior of the final design, the optimal response points are inserted into the FEM, and then it updated and resolved with this new design point. The new solution results are summarized in Figs. 9, 10 as a comparison with the original one. The results carried out indicated that the value of TD after optimization is comparable to its value in Table 2; hence the FEM has correctly updated the optimal design point.

Table 2The initial and optimal design for both screening and MOGA methods

Design variables	Initial design	An optimal candidate design points						Final design
		Screening method			MOGA method
$Z_1$ [mm]	42	50.065	47.125	54.895	41.93	54.658	51.8	51.8
$Z_2$ [mm]	42	59.905	56.204	55.118	58.97	58.655	53.9	53.9
$Z_3$ [mm]	54	58.803	56.782	50.307	58.09	59.050	57.8	57.8
$Z_4$ [mm]	50	43.649	40.078	43.361	32.92	42.377	41.7	41.7
$Z_5$ [mm]	50	13.678	46.713	28.913	20.50	49.146	38.6	38.6
$Z_6$ [mm]	50	38.695	42.194	50.140	32.31	24.923	56.3	56.3
$Z_7$ [kg.m-3]	7850	7759.1	7714.5	7335.2	7540.8	7846.9	7601.4	7601.4
$Z_8$ [GPa]	210	207	204	198	201	200	199	199
$Z_9$ [N]	3700	3677.6	3960.1	3149.7	3551.2	3418.56	3670.8	3670.8
$Z_{10}$ [kg]	6	1.4046	0.3641	1.6882	4.15	1.01	3.53	3.53
Response	–	–	–	–	–	–	–	–
TD [mm]	0.051	0.0654	0.0665	0.0651	0.0661	0.065	0.0650	0.0446
SW [kg]	19.063	15.1	16.06	15.58	15.32	15.26	15.86	15.86

Also, it can be seen from Fig. 9(b), the maximum TD observed is 0.0446 mm at motor rotor position; which it improved to 13.7 % less than the original value shown in Fig. 9(a). Thus, the minimal weight design technique, not only satisfies saving in total material resource consumed, but also the vibration response characteristics can be improved. It confirmed the effectiveness of this method to come across the energy efficient design requirements in terms of energy consumption and material resources used and the waste and pollution created in the process:

1

$\begin{array}{l}\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{d}Z=\left(Z_1,Z_2,Z_3,\dots ,\dots ,Z_p\right),\\ \mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}SW={\sum }_{k=1}^N{\rho }_k{{\rm A}}_{k}L,\\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}\left\{\begin{array}{l}TD\le 0.067\mathrm{m}\mathrm{m},\\ 41\le Z_1\le 55,41\le Z_2\le 60,\\ 54\le Z_3\le 60,10\le Z_4\le 70,\\ 10\le Z_5\le 60,20\le Z_6\le 60,\\ 7200\le Z_7\le 7860,190\le Z_8\le 210,\\ 600\le Z_9\le 4000,0\le Z_{10}\le 10.\end{array}\right.\end{array}$

Fig. 9The vibration mode with unbalance effect: a) initial design, b) optimal design

a)

b)

Fig. 10Comparing the vibration mode shape of the spindle at motor-rotor location