Linear forced-rotordynamics analysis for optimizing the performance factors of machine motorized spindle using design explorer method

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 skew-symmetric 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 linear-equation solvers [4]. In this study, direct integration method was used to find out the steady-state Frequency-Response-Function (FRF) through using ANSYS software.

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 Design-Variables (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 Max-Deformation (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, $\pm 0.25x_1$, $\pm 0.25x_2$, $\pm 0.18x_3$, $\pm 0.05x_4$, $\pm 0.054x_5$, $\pm 0.2x_6$, $\pm 0.25x_7$.

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 observed-values. 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 motor-rotor seat inner diameter ($x_1$) and material density ($x_5$), and rotating unbalance-mass ($x_7$) 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 max-deformation

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 RS-based optimization methods, such as Screening and Malti-Objective-Genetic-Algorithm (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 candidates-points. 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 candidates-point 1, and 2, since its zero in candidates-points 3. The vibration-behavior of optimum FEMs were obtained by creating and updating (real calculation) the DVs of candidates-points 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 vibration-response calculated is at motor-rotor rotating force position, while it minimum at front bearings position. Based on the smallest percentage variant detailed in Table 2, the candidates-point 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 structural-weight 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.

Fig. 4The FEMs mode shapes after optimization

a) First candidates-point

b) Second candidates-point

c) Third candidates-point

d) DO-based 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 Max-Deformation (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.