An improved RCSA for identifying the spindle-holder taper joint dynamics

2.1. Receptance coupling substructure analysis (RCSA)

The ultimate objective of RCSA is to formulate the assembly dynamic response at any spatial coordinate by using the experimental or analytical FRFs of individual substructures. Take the spindle-holder assembly (component $A$) depicted in Fig. 1 as an example. Spindle substructure and holder structure are denoted as $S1$ and $S2$, respectively.

As the axial degree of freedom of holder is constrained due to the clamping force, the translational displacement in $y$ direction can be neglected. However, for the holder bending inside the spindle taper hole, the rotational degree of freedom of holder must be considered. Therefore, as shown in Fig. 1, each reference point has two degrees of freedom (DOF) of $x$ and $\theta$, namely the lateral displacement and the rotational angle, respectively. Thus the corresponding loads will be lateral force $f$ and moment $M$.

As for $S1$, relationship of its receptance matrix $R^{S1}$, response vector $U^{S1}$, and corresponding load vector $F^{S1}$ can be written as:

where:

Fig. 1Spindle-holder assembly and substructures

Here, $R^{S1}$ is a symmetric matrix ($H_{jk}^{S1}=H_{kj}^{S1}$) since the substructure is assumed linear. According to the discussion above about the DOF of each reference point, for arbitrary reference points $j$ and $k$ ($j$, $k=$1, 2,..., $n$) we can write the full FRF matrix as follows:

and

where, $x_j^{S1}$ and ${\theta }_j^{S1}$ are the lateral displacement and rotational angle of point $j$, respectively. $f_k^{S1}$ and $M_k^{S1}$ represent the lateral force and moment applied on point $k$, respectively. Direct ($h$, $P$) and cross ($l$, $N$) FRF elements of the sub-matrix $H_{jk}^{S1}$ are defined as:

According to the reciprocity [20], $N$ can be set equal to $l$, that is:

Similarly, for substructure $S2$ and assembly $A$, Eq. (1) will become:

where:

Also, $R^{S2}$ and $R^A$ are symmetric. And sub-matrix $H_{jk}^{S2}$ ($j$, $k=$1, 2,..., $n$,..., $n+m$) and $H_{jk}^A$ ($j$, $k=$ 1, 2,..., $m$) have the same expression with Eq. (5).

In order to predigest the formula expression, reference points are divided into contact points (1~$n$) and non-contact ones ($n+1$~$n+m$), denoted with the lower script $c$ (contact points) and $i$ (internal points), respectively. Therefore, Eq. (1), Eq. (13) and Eq. (14) can be rewritten as:

On the basis of RCSA theories [8], the load equilibrium and displacement compatibility equations can be presented as:

Combining Eq. (21)-Eq. (25), we can get:

that is:

Substituting $F_c^{S2}$ from Eq. (22) to Eq. (27) yields:

Therefore assembly $A$ receptance matrix $R_{ii}^A$ in terms of the individual substructures receptance matrices is given as:

Eq. (29) is the essential equation of RCSA. It is worth emphasizing that when considering the rotational dynamics of taper joints, all the sub-matrices in Eq. (29) are described in the full form described in Eq. (5).

2.2. The improvement of RCSA

Generally, the assembly FRF obtained though RCSA is used for evaluate the limiting stable depth of cutting which is determined by the formula as follows:

where $K_s$, $\mu$ and $z$ are constant coefficients. It is easy to see that for calculating $b_{critical}$, all we need is the real part of direct displacement/force FRF at the assembly free end, $h\left(\omega \right)$. The other three rotational FRFs ($l$, $N$, and $P$) are unnecessary to require.

However, it is now well known that the joint rotational response is significant for achieving accurate RCSA. In other words, although there is no need to calculate the rotational FRFs at the assembly free end, the rotational dynamics of joint must be considered.

Due to the experiment and instrument conditions, it is quite difficult to measure the rotational dynamics. Some researchers used translational FRF, obtained experimentally, to form the full receptance matrix for RCSA. For instance in [10], as proposed by Schmitz, making use of the experimental data, they applied a second-order backward finite difference method to estimate the rotational FRF. The works presented by Park et al. in [21] proposed an algorithm to extract the joint rotational dynamics from experimental data. Also, a new methodology to estimate the rotational dynamics is described by P. Albertelli et al. in [22]. Their methodology is deduced by taking inspiration from works reported in [10]. However, the main drawback of these proposed approaches is that experimental data is frequently contaminated by environmental noise resulting in the giant computation error.

Hence, aiming at the purpose of solving the rotational FRF issue efficiently and rapidly without sacrificing the accuracy of RCSA, an improved RCSA is proposed in this section. The improvement lies in the modeling approach of joint dynamics. Previously, to model the full dynamic response of joint, including the translational and rotational response, both translational and rotational spring elements were used as a combination. But here, instead, parallel of translational spring elements are used to characterize the full joint dynamics. While this equivalent model is applied in RCSA, relationship illustrated in Eq. (29) is maintained, but only the displacement/lateral force translational FRF, $h$, is used.

In order to verify the equivalent model proposed above, series of numerical simulations are performed. In detail, based on analytical beam theories, three different joint models are established as depicted in Fig. 2. Each model is composed of a hollow cylinder (Beam A, representing the spindle) and a solid cylinder (Beam B, representing the holder) connected through spring elements (Spring, representing the spindle-holder taper joint) to the hollow cylinder. Model I is mostly employed in previous works. In this model, both translational and rotational spring elements at single point are suggested to simulate the joint dynamics. In model II, two points along the interface with parallel translational spring elements are used to simulate the joint dynamics. And model III only has translational spring elements at one point for joint dynamics simulation.

Fig. 2Different joint model

It can be verified that model I is equivalent to model II. Although no rotational spring elements are contained in model II, the joint rotational dynamic behaviors are considered automatically. On the other hand, due to the absence of rotational spring element at one point, model III may not fully reflect the same dynamic behaviors as model I. So model III may lead to the low accuracy of RCSA.

By virtue of Matlab software, simulation has been carried out based on the finite element analysis (FEA). As depicted in Fig. 2, 2-node Timoshenko beam element with 2 DOFs ($x$, $\theta$) at each node (represented by the black dots) is used to establish the beam models. Geometry parameters for the beam model are listed in Table 1. Stiffness coefficients of translational and rotational spring elements are set to equal 5×10⁵ N/m and 3×10⁴ N·m/rad, respectively. Material properties, like density, Poisson’s ratio and modulus, of the beams are 7850 kg/m³, 0.3 and 2.06×10¹¹ Pa, respectively. The corresponding eigenvalue problems then are solved to extract the first five eigenvalues (natural frequencies) and eigenvectors (mode shapes).

Mode shapes and the corresponding natural frequencies are described in Fig. 3. Table 2 resumes the errors in the estimation of natural frequencies extracted from the three models. Comparing with the results extracted from model I, calculation accuracy of model II is acceptable for the first mode, and excellent for the rest four. However on the contrary, results extracted from model III deviate from those of model I considerably. Model III simply looses the 1st mode (46 Hz), and frequency deviations of the 2nd and 4th modes are 34.02 % and 18.71 %, respectively.

It is therefore a fact that translational and rotational dynamics of joint can be fully reflected by using a parallel translational spring elements at multiple points along the joint interface (Model II). In this case, we can conclude that if the translational FRF at the free end of assembly structure is required, i.e. for predicting the cutting stability lobe diagram, one can only use the joint translational FRF for RCSA while the rotational dynamics is already considered automatically. The improved RCSA illustrated in this section is less time-consuming, where the RCSA accuracy is still guaranteed regardless the avoidance of using joint rotational FRF ($l$, $N$, $P$).

Fig. 3Extracted eigenvalues and eigenvectors (solid-Beam A, dash-Beam B)

a) Model I

b) Model II

c) Model III

3.1. Definition of spindle-holder taper joint dynamic behavior

The fundamental formulation for identification of spindle-holder taper joint dynamic behaviors is Eq. (22). The goal is to identify the joint receptance matrix $R_{cc}^{S1}$. The receptance matrices, i.e. $R_{ii}^{S2}$, $R_{ic}^{S2}$, $R_{cc}^{S2}$ and $R_{ci}^{S2}$ of the holder (substructure $S2$) can be obtained using beam theory and FEA, since the holder is a linear structure. Receptance matrix $R_{ii}^A$ of assembly $A$ is measured experimentally by impact testing. Assume that:

yields:

Once $G$ is provided, spindle-holder taper joint receptance matrix $R_{cc}^{S1}$ can be calculated directly from Eq. (32).

Again we must emphasize the idea that receptance matrix $R_{cc}^{S1}$ represents the dynamic behavior of taper joint and spindle itself simultaneously. The matrix $R_{cc}^{S1}$ here is different from that obtained from experiment performed on the spindle. In fact, the taper joint inside substructure $S1$ is inaccessible to perform impact test. As long as the spindle-holder taper joint has the same conditions, i.e. geometry information and clamping force, the taper joint receptance matrix can be regarded as the constant property of spindle-holder assembly.

3.2. Dimensions of spindle-holder taper joint receptance matrix

Dimensions of $R_{cc}^{S1}$ depends on the number of elements required identification. According to Eq. (2) and Eq. (5), $R_{cc}^{S1}$ is a 2$n$×2$n$ matrix due to the 2 DOFs ($x$, $\theta$) at each reference point. Considering the symmetry of $R_{cc}^{S1}$ and reciprocity of its sub-matrix, as Eq. (12), there are $3\times \left[\right(1+n)\times n/2]$ independent elements. Among these independent elements, $\left[\right(1+n)\times n/2]$ of them are displacement/force translational FRFs, while the rest are rotational FRFs. Based on improve RCSA methodology proposed in Section 2.2, parallel translational FRFs of joint can express both its translational and rotational dynamics. Thus, the number of elements required identification in $R_{cc}^{S1}$ is $\left[\right(1+n)\times n/2]$.

On the other hand, for determining $R_{cc}^{S1}$, $R_{ii}^A$ ought to at least have the same number of independent measured elements. The number of independent measured elements in $R_{ii}^A$ is $\left[\right(1+m)\times m/2]$, as we can only measure the translational FRF. For sake of simplicity, we assumed that:

Moreover, based on [23], we can prove that $R_{ii}^A$ actually contains only 3 independent elements, while the other FRFs can obtained directly though RCSA. So $R_{cc}^{S1}$ has 3 independent elements in accordance with Eq. (32).

Hence, we have:

then the number of reference points, which is finally depicted in Fig. 4, is obtained:

It can be concluded that spindle-holder taper joint receptance matrix $R_{cc}^{S1}$ is a $2\times 2$matrix. In addition, the improved RCSA in Section 2.2 has validated that a couple of parallel translational FRFs at 2 points along joint is sufficient for predicting the translational FRF at free end of assembly structure.

Fig. 4Reference points on spindle-holder assembly and substructures

3.3. Spindle-holder taper joint receptance matrix calculation

Consequently, Eq. (21)-(23) can be rewritten as:

Assume that $g_1$, $g_2$ and $g_3$ are the three independent elements of matrix $G$ in Eq. (31):

Eq. (29) can be updated:

where:

Once $g_1$, $g_2$ and $g_3$ are known, Eq. (32) can be used to identify spindle-holder taper joint receptance matrix $R_{cc}^{S1}$. The proposed identification technique is implemented applying the following steps which are also illustrated in Fig. 5:

1) Impact test on spindle-holder assembly $A_1$.

2) FEA for holder $S{2}_1$ in free configuration.

3) Identification of spindle-holder taper joint receptance matrix $R_{cc}^{S1}$ using improved RCSA.

4) FEA for holder $S{2}_2$ in free configuration.

5) Prediction of assembly FRF $h$ at the free end (point 2) of $A_2$ based on improved RCSA.

6) Impact test on spindle-holder assembly $A_2$ to validate the prediction performed in Eq. (5).

Fig. 5Complete process for the identification and validation of taper joint dynamics

4.1. Acquisition for the receptance matrix of holder substructure

Since the Euler-Bernoulli beam theory neglects the effects of rotary inertia and shear deformation, it has been found insufficient for structure dynamic modeling [24-27]. Timoshenko beam theory is employed to model the BT 50 holder substructure (see Fig. 6(a)) in this paper. Proportional or Rayleigh is assumed for structure dynamic modeling, thus we have:

where $\alpha$ and $\beta$ are the mass and stiffness damping coefficients, respectively. Both coefficients can be obtained from impact test [28]. Then based on the principle of Hamilton and FEA, general dynamic equation for holder substructure is written. See Eq. (42):

where $\left[\mathbf{M}\right]$, $\left[\mathbf{C}\right]$ and $\left[\mathbf{K}\right]$ are mass, damping and stiffness matrices, respectively. $\left\{\mathbf{U}\right\}$ is the displacement vector. $\left\{\mathbf{F}\right\}$ is the force vector.

Dynamic model of holder $S{2}_1$ is established. The elastic modulus and density of holder are 2.06×10¹¹ Pa and 7850 kg/m³, respectively. The model is validated experimentally through impact test. Experimentation setup is shown in Fig. 6, where single direction accelerometer is used for measuring the response. The simulated and measured direct FRF at each reference point of holder $S{2}_1$ are compared in Fig. 7, where the agreements are excellent. Hence the analytical approach for dynamic modeling substructure $S{2}_1$ is reliable. The $4\times 4$ matrix in Eq. (37) is then obtained numerically based on this dynamic model.

Fig. 6Impact test on holder S21 in free configuration

Fig. 7Comparison of each direct FRF at the reference points of holder S21

4.2. Acquisition for the receptance matrix of spindle-holder assembly

In this section, substructure $S{2}_1$ is mounted on a machine tool spindle system to form the spindle-holder assembly $A_1$. Impact tests are performed on the overhang as shown in Fig. 8, where point 2 is the free end of $A_1$. The 2×2 matrix in Eq. (38) is then obtained experimentally.

Fig. 8Impact test on spindle-holder assembly A1

During the measuring procedure, one must be aware of the coherence of the input and output signal. Coherence simply reflects the signal quality. As depicted in Fig. 9, within the frequency range (0-2000 Hz) we concern about, most part of the coherences of estimated FRF approximate to 1. Coherence of $h_{11}$, direct FRF at point 1, is not good close to 1120 Hz. One possible cause may be the characteristics of accelerometer [29]; another one may be the nonlinear feature of the machine tool due to the joints like bearings, ball screws, bolts and so on. But in general, the experimental data are reliable.

Fig. 9Coherences of impact test

4.3. Identification for the spindle-holder taper joint receptance matrix

Substituting the matrices obtained in Section 4.1 and 4.2 into Eq. (41) and Eq. (32), the spindle-holder taper joint receptance matrix $R_{cc}^{S1}$ is finally determined. In this section, two more different holders, $S{2}_2$ and $S{2}_3$, are used to identify $R_{cc}^{S1}$ by repeating the works in Section 4.1 and 4.2. Geometry information can be found in Fig. 10 and Table 3. Three holders have different dimensions of overhang, but the same taper end geometry and clamping force. For convenience, taper joint receptance matrices obtained by using three different holders are denoted as ${}_{}{}^{S{2}_1}{R}_{cc}^{S1}$, ${}_{}{}^{S{2}_2}{R}_{cc}^{S1}$ and ${}_{}{}^{S{2}_3}{R}_{cc}^{S1}$, respectively.

Direct ($h_{11}^{S1}$, $h_{22}^{S1}$) and cross ($h_{12}^{S1}$) FRF elements in ${}_{}{}^{S{2}_1}{R}_{cc}^{S1}$, ${}_{}{}^{S{2}_2}{R}_{cc}^{S1}$ and ${}_{}{}^{S{2}_3}{R}_{cc}^{S1}$ are compared in Fig. 11.

Fig. 10Three holders with different overhang dimensions

Fig. 11Elements in taper joint receptance matrix RccS1 S21, RccS1 S22 and RccS1 S23

a) $h_{11}^{S1}$

b) $h_{12}^{S1}$ (or $h_{21}^{S1}$)

c) $h_{22}^{S1}$

Fig. 12Holder and bolt nail

Fig. 11 shows good agreements among curves. However, differences do exist among curves, especially in a). The reason is that when experiments are carried out on assemblies ($A_1$, $A_2$ and $A_3$) with respect to three different holders, same clamping conditions are not guaranteed strictly. As shown in Fig. 12, the holder is clamped through a bolt nail which is mounted on the holder taper end manually. Workers simply cannot assemble this bolt nail with the same torque every time, which, we believe, results in ${}_{}{}^{S{2}_1}{R}_{cc}^{S1}$, ${}_{}{}^{S{2}_2}{R}_{cc}^{S1}$ and ${}_{}{}^{S{2}_3}{R}_{cc}^{S1}$ the different clamping conditions.

But in general, matches with each other well. This observation helps to validate the conclusion that dimensions of holder overhang have no effect on the taper joint dynamic behaviors.