Study of experimental modal analysis method of machine tool spindle system

2.1. PloyMAX algorithm

The least-squares complex frequency-domain (LSCF) method identifies a common-denominator model and was introduced to find initial values for the iterative maximum likelihood method. These “initial values” yielded very accurate modal parameters with a very small computational effort and provide very clear stabilization diagrams. The PolyMAX method is a further evolution of the LSCF estimation method. Just like the FDPI (Frequency-domain direct parameter identification) method, PolyMAX uses measured FRFs as primary data. In the PolyMAX method, following so-called right matrix-fraction model is assumed to represent the measured FRFs:

where $\left[H\left(\omega \right)\right]\in C^{N_0\times N_i}$ is the matrix containing the FRFs between all $N_i$ inputs and all $N_0$ outputs; $\left[{\beta }_r\right]\in C^{N_0\times N_i}$ are the numerator matrix polynomial coefficients; $\left[{\alpha }_r\right]\in C^{N_i\times N_i}$ are the denominator matrix polynomial coefficients and $n$ is the model order. A so-called $z$-domain model is used in Eq. (1) with: $Z=e^{-j\omega ∆t}$, $\mathrm{\Delta }t$ is the sampling time. Eq. (1) can be written down for all values $\omega$ of the frequency axis of the FRF data. Basically, the unknown model coefficients $\left[{\beta }_r\right]$ and $\left[{\alpha }_r\right]$ are then found as the least-squares solution of these equations. Once the denominator coefficients $\left[{\alpha }_r\right]$ are determined, the poles and modal participation factors are retrieved as the eigenvalues and eigenvectors of their companion matrix:

The poles $e^{-{\lambda }_i∆t}$ are in the diagonal matrix $\mathrm{\Lambda }$, ${\lambda }_i$ with eigenfrequencies ${\omega }_i$ and damping ratios ${\epsilon }_i$ as follows:

2.2. Complex mode indicator function (CMIF)

The CMIF approach uses a decomposition of the FRF matrix to determine the number of references necessary in order to obtain an adequate experimental modal model. Using only this limited set of data, subsets of references of this FRF matrix are evaluated to determine suitable locations for the references prior to actually performing the modal survey [13]. The FRF matrix is defined as follow:

where $\left[H\left(\omega \right)\right]\in C^{N_0\times N_i}$ and $\left[\varnothing \right]\in C^{N_0\times 2N}$ when $N$ degrees of freedom system has 2$N$ conjugate complex roots, ${\lambda }_i$ is the pole of the modal, ${\left[L\right]}^T\in C^{2N\times N_0}$ is the inverted of modal participation factor matrix. The Complex Mode Indicator Function is defined as the eigenvalues solved from the normal FRF matrix at each spectral line and is normally expressed using Singular Value Decomposition notation as:

In which [$H$] is the FRF matrix; [$U$] is the left singular matrix corresponding to the matrix of mode shapes; [$S$] is the diagonal singular value matrix; [$V$] is the right singular matrix corresponding to the matrix of modal participations. To any modal of the system, frequencies of local singular value peaks indicate damped natural frequencies, and using the regular matrix as:

It comes with the CMIFs:

where ${\mu }_k\left(\omega \right)$ is eigenvalue of the regular matrix of FRFs, $s_k\left(\omega \right)$ is eigenvalue of the matrix of FRFs, $k$ is the freedom from 1 to $N_i$. At the resonance of a mode, at least one CMIF will peak; repeated modes would be indicated by two or more CMIFs peaking at the same frequency.

2.3. Modal assurance criterion (MAC)

One of the most popular tools for the quantitative comparison of modal vectors is the Modal Assurance Criterion (MAC) [14]. It is bounded between 0 and 1, with 0 representing no consistent correspondence and 1 indicating fully consistent mode shapes. The MAC matrix is defined as:

where ${\psi }_i$ and ${\psi }_j$ are two mode shape vectors of mode $i$ and $j$. To ensure the linear independence of the tested modal vectors, the minimum off diagonal terms modal assurance criterion is considered as an effective method for optimal sensor placement [15].

3.1. Experimental setup

The sensor including the reference and response determines the accuracy of the testing data and the modal analysis results of the spindle system. To study the topic, the experiment is setup as shown in the Fig. 1. The impact modal test is applied on the spindle system with two methods of hammer and accelerometer as reference.

3.2. Reference number of the experiment

Basically there is no difference between a hammer reference and accelerometer reference modal test, which is the one row or column of the FRF matrix in modal test. But there are some very basic practical aspects that may cause some differences, so the modal analysis results in admissible difference of two methods. Normally, multiple reference measurements reduce the likelihood of “missing” a mode during the curve fitting process [17]. Multiple reference curve fitting offers advantages when the FRF data contains closely coupled modes or truly repeated modes. However it is depends on the theoretical analysis of general structures. To some cases like the spindle systems, what really important is the error sensitive modes. Furthermore, the multiple references may reflect on the redundancy of modal information and results in the selection error of the modal parameters.

Fig. 1Experimental setup

Fig. 2Stabilization diagram of different reference number: hammer-reference modal analysis

a) Three reference points $X$\$Y$\$Z$

b) Two reference points $X$\$Y$

c) One reference points $X$

Fig. 3Stabilization diagram of different reference number: accelerometer-reference modal analysis

a) Three reference points $X$\$Y$\$Z$

b) Two reference points $X$\$Y$

c) One reference points $X$

In this paper, the same experiment is done and the frequency range of 50-1280 Hz is applied on the impact test. Then the estimate results of every processing is showed in the stabilization diagram, the symbols have following meaning: ‘$o$’: new pole, ‘$f$’: stable frequency, ‘$d$’: stable frequency and damping, ‘$v$’: stable frequency and eigenvector, ‘$s$’: all criteria stable. Also the figure shows the sum of the FRFs and the modal order in estimation.

By the results shown in Fig. 1 and Fig. 2, it is concluded that whatever the reference is based on hammer or accelerometer, the stabilization diagram indicate that adding the reference quantity will result in instability using the same modal size in a test. It is because the multiple reference points would bring redundant information in data processing, and reflect on the modal parameters calculation. Especially for the different test in moving hammer and accelerometers, the data noise would also bring the modal information deviation.

For some structures, the more reference is selected the easier the modes could be obtained in the modal test. This is because that it is not theoretically possible to find a proper reference degree of freedom (DOF) where all the modes have sufficiently high participation for proper modal model extraction. Furthermore, the multi-reference test is also required in cases where the structure has more modes with the same resonance frequency. However, for the structure like machine tool spindle systems, it is not likely to have local modes and the structure fixed is not certain symmetrical. In the paper, modal parameters are got from the modal estimation with the help of mode indicator functions, FRFs, modal synthesis FRFs and modal validation tools. As indicated in the Tables 1 and 2, the single-reference got more comprehensive information than the multi-reference experiment in the same modal frequency bandwidth and estimation order. It is concluded that the added reference DOF(s) does not add more information about these modes but on the contrary bring the trouble in modal parameter selection with redundant information. Therefore, in impact modal analyses of those spindle systems, it is not simply to add reference DOF(s), but to selection the better position to achieve the concerned modal information.

3.3. Reference direction of the experiment

In order to get the best information of reference DOF, in this paper, the direction section of reference at one position is experimented and analyzed. The position is based on the proper modal extraction position verified at the non-node of the mode. As demonstrate in the Fig. 4 and Fig. 5, the single-reference tests of different reference directions are conducted with both hammer and accelerometer referenced methods. By contrast, the $X$-direction stabilization diagram contains more modal information than others. And the $Z$-direction results in very different graph by the effect of some possible spurious modes.

Fig. 4Stabilization diagram of different reference directions: hammer-reference modal analysis

a) Reference points $X$ direction

b) Reference points $Y$ direction

c) Reference points $Z$ direction

Fig. 5Stabilization diagram of different reference directions: accelerometer-reference modal analysis

a) Reference points $X$ direction

b) Reference points $Y$ direction

c) Reference points $Z$ direction

To further study the modal analysis parameter results in Table 3 and Table 4, the $X$-direction results have the most integrated of the spindle. And the $Y$-direction would lose some mode, $Z$-direction do similarly but give prominence to the $Z$-direction mode which is less important in this case. When analyzing the spindle structure which is fixed on the machine tool in $Z$-direction, it is discovered that the $X$-direction have the most modes, the empirical analysis matches the experimental results.

In cylindrical grinding, the radial and tangential vibration is more important than that of axial. Therefore, the modes of the $X$ and $Y$ directions occupy the dominant position in the research, and the proper modal extraction should be in these error-sensitive directions. Based on the study, it is concluded that the error-sensitive modes is important to analysis the spindle systems rather than the global ones. Furthermore, the error-sensitive modes are effective in experiment-based model updating.

4.1. Finite element method result

To see the reliability of the modal analysis results, a finite element model is developed to assistant observe the modal parameters of modal frequency and shape. The FE model of spindle system is modeled according to the followings:

a) Some small features which would not affect the system modal stiffness and modal mass, for example small holes, shaft shoulders and angle of chamfers are neglected in this model.

b) The mesh type of global system is hexahedron 20.

c) The bolt and screw property are specified as the property of the bar element.

d) The clearance fit property in the spindle is modeled as a contact connection with negative clearance value and both parts are linked together in the tangential directions. The fitting parts conclude the outer ring with the bearing support, the inner ring with the shaft, the stator with the support and the rotor with the shaft.

e) As bearings, the spider element is applied, with one center node represents the inner ring of bearing. The outer ring is connected with it by treating the balls of bearing as springs in $X$, $Y$ and $Z$ directions.

f) Other small parts are assumed to be completely constraint to the main parts.

g) The bottom of the spindle system is regard as DOFs restraint to the ground to simulate the fixing situation in the machine tool.

The material of the shaft is 42 CrMo, the rotor and stator is composed of silicon sheet and copper, 45 steel are respectively used for spindle box, bearings and other parts. The spindle system is fixed on the ground to simulate the condition that the spindle box is installed on the machine. The model, with more than 250000 nodes is depicted in Fig. 6 and dynamic characteristics of spindle system were then studied using Nastran software. A total of 14 mode shapes and natural frequencies were extracted in the 0-1280 Hz frequency range. The comparison between the EMA and FEM results are shown in Table 5. Expect for the 3 grinding wheel modes, the other 11 modes of EMA ($X$ direction single reference) and FEM are in acceptable agreement, which in other way indicate the $X$ direction single reference impact test have the reliable analysis of this system. In this paper, the FEM is not the topic issue, so there is no more discuss about the modeling error.

4.2. CMIFs

Due to the shortages of FEM, the verification is applied using the CMIFs. The CMIF will exhibit frequencies of local peaks to indicate damped natural frequencies. Depending on the number of references, higher order CMIFs can be computed to determine the multiplicity of the repeated root. To different references, if the added reference would influence the local peaks number. If the number of peaks observed in the CMIFs plot is unchanged, then these reference locations are likely to not be needed to adequately excite all the modes. On the other way, if the number changed, then these reference locations are needed to adequately excite all the modes. Therefore, the CMIFs plot can determine the number of references necessary in order to obtain an adequate experimental modal model.

Fig. 6Finite element model of the spindle system

Fig. 7CIMFs of different reference points

As displayed in the Fig. 7, the four different CMIFs exhibit frequencies of local peaks. From the view of all the frequency range, the $X$-reference and $XYZ$-reference functions have the most peaks. What is more, the peaks of $X$-reference function between 300 to 600 Hz have small amplitude while the $XYZ$-reference function has many noisy peaks during the frequency range. And according to the figure, comparing the $Y$-reference and $Z$-reference with the $X$-reference function, adding the references, the peaks of CMIFs plot is unchanged. What is worse, it will bring more data noise to the system. Therefore, hybrid experimental-numerical study show that the $X$-reference impact modal testing method on the proper extraction position is a feasible and reliable way to achieve the accurate experimental modal parameters.