Efficient joint identification and fluted segment modelling of shrink-fit tool assemblies by updating extended tool models

For identifying the joint parameters between the shrink-fit holder and tool, a carbide blank of the same length as the tool is inserted in the holder with the same insertion depth. In this paper, the holder-tool joint is represented by a spring-damper element with a translational and a rotational degree of freedom. However, the proposed approach can be applied for identifying the parameters of other joint models such has the uniform elastic layer or multiple point joint model. For a shrink-fit joint, a mass-less spring-damper element can be assumed because the masses of the substructures are several orders of magnitude larger than the mass representing the joint. For such an element, the complex stiffness matrix \({\varvec{K}}\) is given by,where \({k}_{xf}\) corresponds to the translational stiffness and \(i{\omega c}_{xf}\) to the equivalent translational frequency-dependent damping. Similarly, \({k}_{\theta yM},{c}_{\theta yM}\) correspond to the rotational stiffness and damping factor. The cross-stiffness and damping terms are ignored for simplicity. The elastic coupling of the extended holder (\(ITh\)) with the carbide blank (\(C\)) is achieved by the following coupling equation and gives the compliance matrix at point 3 of the extended tool assembly \(IThC\):

Considering only the displacement to force FRF,

Since the joint parameters are initially unknown, the displacement to force FRF \({H}_{33}^{IThC}\) is calculated without the stiffness term. The corresponding FRF of the freely constrained tool assembly \({H}_{33,m}^{IThC}\) is obtained using the setup shown in Fig. 2b. The subscript ‘\(m\)’ denotes an experimentally obtained FRF. The influence of sensor mass loading on the measured FRF was compensated using the structural modification method [12]. Figure 3 shows the measured and predicted tip FRFs for a tool holder with two 16 mm carbide blanks with overhangs of 60 mm and 100 mm. The predicted and measured FRFs have similar courses, however, due to the missing joint parameters, the predicted FRF has resonance peaks located at relatively higher frequencies. The joint parameters can now be identified by comparing of both these FRFs. Literature provides several approaches for identifying joint parameters. Some of them are: manual or visual tuning of parameters [13, 21], parameter variation [1] and curve fitting or non-linear optimization [9, 15]. The objective function for these approaches has been to reduce the difference between the predicted and measured amplitudes within a frequency range. For the freely-constrained case, a much better indicator of the influence of the joint stiffness is the location of the resonance peaks. Hence, in the proposed approach, the objective function is defined to reduce the weighted difference between the location (frequency) of the predicted and measured resonance peaks:

where, \({f}_{i}^{m}\) corresponds to the frequency of the \(i\)-th resonance peak of the measurement and \({f}_{i}\) to that of the calculated peak. \({W}_{i}^{f}, {W}_{i}^{abs}\) are the weightages assigned to the \(i\)-th frequency and absolute error respectively. Where, \({W}_{i}^{f}\gg {W}_{i}^{abs}\) to enable a faster convergence of the optimization problem. A genetic algorithm was utilized to solve the optimization problem so that a global minimum to the bounded problem could be found. The identified joint parameters were used to elastically couple the interface-holder assembly with the one segment beam representing the blank (Fig. 3). A very good correspondence can be observed between the measurements and the prediction with identified joint parameters. Both, the poles as well as the zeros could be matched well by the identified joint parameters. Due to the free-free condition, the structural modes of the tool assembly can be observed without the disturbing influence of surrounding structure. In addition, the use of acceleration sensor enables the reliable measurement till higher frequency (16 kHz).

Once the joint properties for the holder-carbide blank have been identified, the carbide blank is replaced with the corresponding real tool with the same insertion length. The extended tool model is also modified by replacing the one-segment beam with the elastic coupling of a two-segment beam. The two segments are intended to represent the overhanging shank part of the real tool (\({T}_{\mathrm{s}}\)) and the fluted segment (\({T}_{\mathrm{f}}\)) respectively (Fig. 1b). Initially, both the beams possesses the same diameter of the shank. The shank segment \({T}_{\mathrm{s}}\) is first coupled elastically using the identified joint properties with the extended holder at point 2 giving the compliance matrix at the free-end of the shank as,

Applying the force equilibrium and compatibility conditions between the shank and the fluted segment at 4, gives the expression for the direct compliance matrix of the complete tool \(T\) at 4 in the rigidly coupled state,

In this formulation, the compliance matrix \({{\varvec{G}}}_{44}^{ITh{T}_{s}}\) is available from Eq. 7 and the reference displacement to force FRF \({H}_{44m}^{IThT}\) is obtained from a measurement of the freely-constrained tool assembly (Fig. 2c). Here also, an acceleration sensor is used to allow reliable measurement till higher frequencies (16 kHz) and its mass loading effect is compensated for. The unknown compliance matrix \({{\varvec{G}}}_{44}^{{T}_{f}}\) in Eq. 8 depends on the geometric properties of the equivalent fluted segment beam. Since the reference FRF is available, the geometric parameter (\({d}_{eff}\)) can be varied until a good correspondence between the measured and predicted FRF is achieved. For this, a linear optimization algorithm, which minimizes an objective function similar to the one suggested in Eq. 6, is selected. The formulation in Eq. 8 is advantageous for this estimation because of the following reasons: (a) out of the four compliance matrices of the fluted segment beam \({T}_{f}\) only a single compliance matrix at 44 is required in the above expression. This greatly improves the efficiency of the optimization algorithm, as only the compliance matrix at one end needs to be calculated. (b) The measurement of \({H}_{44m}^{IThT}\) on the shank is more convenient than tap testing on the free end of the fluted segment. This is because of the difficulty of attaching sensors on the tool edges and the risk of damage to impact hammer. The measured and calculated (full cylinder) \({H}_{44}^{IThT}\) FRFs for two real end-mill tools with tool diameter of 16 mm and overhang lengths of 58 mm (Z = 3) and 100 mm (Z = 4) are shown in Fig. 4. The detailed description of these end mills can be found in Table 2. FEM simulations showed that the bending of the fluted section contributes significantly to the first mode shape. Therefore, the first resonance provides an excellent reference for identifying the equivalent effective diameter. The higher bending modes do not correspond to significant modal strains of the fluted segment alone. Hence, the correspondence of the predicted FRF with \({d}_{eff}\) is much more important at the first resonance than the higher ones. It can be observed that the resonance frequency of the first bending mode for the full cylinders lies at a lower frequency while the other modes lie at higher frequencies compared to the measurement. The higher modal mass without proportional increase in modal stiffness associated with the full cylinder could explain the shift to lower frequency. The effective diameter estimated by the linear optimization algorithm is used to create the second beam of the two-segment beam. As a check, displacement to force FRFs \({H}_{44}^{IThT}\) with the effective diameters are also plotted in Fig. 4. Here as well, a good correspondence between the measured and predicted FRFs can be observed. Especially the first bending mode could be represented accurately. Clearly, the effective diameter is a simplification of the complex tool geometry and cannot capture all dynamic effects corresponding to the higher modes. The following experimental implementation and validation shows that this simplification provides sufficiently accurate prediction without extensive modelling effort.

For the selected Haimer holder, six carbide blanks of 16 mm diameter and varying lengths of 100, 110, 120, 130, 140 and 150 mm were used. Utilizing the maximum insertion length of 50 mm, gives overhang lengths \({L}_{o}\), from 50 to 100 mm. The freely-constrained tool tip FRF \({H}_{33m}^{IThC}\) was measured for each holder-blank combination using the setup shown in Fig. 2b. The holder was suspended using elastic bands. A uniaxial piezoelectric charge accelerometer of type 4374 from Brüel & Kjaer and an impulse hammer from Kistler of type 9724A with a steel tip were used. An extended tool model was created for each holder-blank assembly and the tool tip FRF was first predicted by rigidly coupling the blank with the holder. Table 1 lists the used blanks with the corresponding measured and predicted (without joint parameters) resonance frequencies for the first and second modes. With increasing overhang, the relative difference between the resonance frequencies reduces (Fig. 5b). The results of the joint stiffness identification are also listed in Table 1 and represented graphically in Fig. 5a. Both translational (\({k}_{xf}\)) and rotational (\({k}_{\theta yM}\)) stiffness show a decreasing trend with increasing overhang. Between the minimum and maximum overhang, the identified translational stiffness reduces by 50.5% and rotational stiffness declines by 25%, indicating that the location of the resonance frequency of the assembly is more sensitive to change in rotational than in translational stiffness. The decreasing trend corroborates well with the decreasing relative errors between the predicted (without \({\varvec{K}}\)) and measured resonance frequencies because a smaller magnitude of additional stiffness is required to compensate for a small difference in resonance frequencies. However, for the same insertion length, there is no physical change in the joint when the overhang is increased. Other complex joint models that consider the variation of contact pressure and stiffness along insertion length like the continuous uniform distributed spring-damper element model by Rezaei et. al [16] or the multipoint RSCA joint model by Schmitz et. al [21] do not show a sensitivity to overhang length. These models provide a more realistic representation. The variation of joint parameters for the same insertion length is probably due to the composite beam lumped spring-damper joint model, where the contact between the inserted tool and the holder is assumed to be rigid and the joint compliance is assumed to be acting at the interface between the holder and the overhanging tool. However this model can still accurately depict the joint behavior at different overhangs and is chosen in this work because of its simplicity and because the focus is on the method of identification. As mentioned previously, the proposed freely constrained approach can also be implemented to parameterize other joint models. The bounds for the optimization of stiffness terms were chosen to be \({k}_{xf}\in [1\mathrm{E}7\;1\mathrm{E}10]\) and \({k}_{\theta yM}\in [1\mathrm{E}4\;1\mathrm{E}7]\).

The identification of joint damping factor \({c}_{xf}, {c}_{\theta yM}\) in the freely constrained state still remains a challenge. It was observed that the amplitude of the resonance frequencies in this state is extremely sensitive to the quality of impact and measurement setup such that a consistent estimation of joint damping was not possible. Hence, literature values for joint damping [13] were used as starting values in this study and adjusted manually. Taking these values and the identified joint stiffness, the one-segment beam was elastically coupled with the holder. The holder-tool was subsequently coupled with the spindle-side compliance matrix to obtain the displacement to force tool-tip FRFs \({H}_{33}^{SThC}\) in the clamped state in X-direction. For validation, corresponding reference FRFs \({H}_{33m}^{SThC}\) were measured with the help of a single-beam laser Doppler vibrometer and the impulse hammer (Fig. 6).

Figure 7 shows the comparison of the amplitude and phase diagrams for the predicted and measured tip FRFs. A very good correspondence could be achieved between the two for all overhang lengths indicating an appropriate identification of the joint stiffness. The adjusted damping is also listed in Table 1. Only for the 50 mm and 60 mm overhang, a certain amplitude error is observed in the clamped-free state. Since the free-free comparison for these tools show a good fit (Fig. 3), it can be assumed that the joint parameters could be identified correctly. The cause for the deviation is probably the composite beam and lumped spring-damper model, which perhaps cannot alone compensate for the deviation between rigidly coupled and measured free-free behavior for short tools.

Three carbide end mills with different lengths, diameters, number of teeth, fluted lengths and helix angles were chosen for the validation of the estimation of the effective diameter. The details of the used holders and tools are listed in Table 2. The joint identification for these tool lengths and holders was carried out with carbide blanks as explained in Sect. 2.1. Effective diameter of the fluted segment for the each tool was estimated based on the measured FRF \({H}_{44m}^{IThT}\) of the extended tool assembly as shown in Sect. 2.2. The analysis of each end mill is done on a case-by-case basis.

A long four-fluted end mill with 16 mm diameter with a total length of 150 mm was inserted in the same shrink-fit holder with full insertion. The joint parameters of the carbide tool with the same total length were used for the tool model. An effective diameter of 12.4 mm was estimated using the freely constrained FRFs shown in Fig. 4b. The comparison of the predicted and measured tool tip FRF \({H}_{33}^{SThT}\) is provided in Fig. 8. Although a certain deviation is observed at around 1120 Hz, the dominant tool mode as well as the quasi-static stiffness could be estimated with sufficient accuracy. A rough estimation of these aspects without extensive modelling is sufficient for predicting the surface location error due to tool deflection and for avoiding excitation of dominant resonance frequencies through process excitation.

As further validation, a Haimer tool holder with 10 mm inner diameter and a two-fluted rounded end mill of total length of 108 mm was taken. The holder was modelled analytically using six Timoshenko beam segments. The results of the joint identification with a fully inserted carbide blank are listed in Table 3. Compared to the 16 mm holder this joint has roughly a factor 10 lower stiffness values. The asymmetric geometry of two-fluted segment results in variation of dynamics with angle of rotation of the end mill. However, for this tool no significant variation in dynamics was observed. This can be attributed to the shorter fluted length compared to the overhang length and corresponds to the observations in [13]. The comparison of the predicted and measured tool tip FRF \({H}_{33}^{SThT}\) shows that the tool mode could be predicted well. Considering the fluted segment as a full cylinder resulted in a poor prediction for all end mills.