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Automated machine tool dynamics identification for predicting milling stability charts in industrial applications

Automated machine tool dynamics identification for predicting milling stability charts in... As the machine tool dynamics at the tooltip is a crucial input for chatter prediction, obtaining these dynamics for industrial applications is neither feasible through experimental impact testing for numerous tool-holder-spindle combinations nor feasible through physics-based modeling of the entire machine tool due to their sophisticated complexities and calibrations. Hence, the often-chosen path is a mathematical coupling of experimentally measured machine tool dynamics to model-predicted tool-holder dynamics. This paper introduces a novel measurement device for the experimental characterization of machine tool dynamics. The device can be simply mounted to the spindle flange to automatically capture the corresponding dynamics at the machine tool side, eliminating the need for expertise and time-consuming setup efforts thus presenting a viable solution for industries. The effectiveness of this method is evaluated against conventional spindle receptance measurement attempts using impact tests. The obtained results are further validated in the prediction of tooltip dynamics and stability boundaries. Keywords Chatter stability · Machine tool dynamics · Piezoelectric actuators · Structural coupling · Inverse RCSA 1 Introduction Regardless of the stability models being formulated in frequency [2–4] or time domain [5, 6], the generation of Until today, regenerative chatter has been the main factor SLDs requires precise knowledge of cutting forces and the limiting the productivity of milling operations in manufac- dynamics at the tip of end mills, where the cutting forces turing industries. Despite the extensive research on this topic excite the machine tool structure. While the cutting forces since its first attention by Tlusty and Polacek [1] in 1963, a can be characterized by simplified coefficients [2], the com- solid solution implementable in an industrial environment is plexity of obtaining machine tool dynamics has remained the still lacking. major obstacle in industrial profits from SLD predictions. It While stability lobe diagrams (SLDs) provide a powerful is worth noting that the knowledge of system dynamics is solution to predict the boundary between stable and unstable not merely useful in SLD creation, but also essential for con- axial depth of cut as a function of spindle speed [2], they are trol applications, where the interaction between mechanical barely used in industries due to the significant measurement and mechatronic subsystems determines the performance of and modeling effort that is required for their creation. machine tool controllers [7, 8]. Further insights into applica- An existing SLD allows for a proper selection of depth tions of machine dynamics can be found in [9]. of cut and spindle speed leading to a significant productivity Modeling the entire machine tool structure including the increase while chatter and its negative consequences are pre- tool-holder assembly is challenging, due to missing informa- vented. However, a blind selection of process parameters may tion (such as the bearing locations) and the unknown damping lead to poor surface quality, significant tool wear, damage to characteristics of the numerous joints of the machine tool the machine tools, increased production cost, and excessive structure [10, 11]. Hence, a direct experimental measurement noise emissions. of the tooltip dynamics using impact hammers may seem an immediate option. While direct measurement of tooltip dynamics can be used for the generation of a precise SLD, B Vahid Ostad Ali Akbari it can be very time-consuming and prone to errors due to [email protected] inaccurate impact location and direction during manual tap Institute of Machine Tools and Manufacturing (IWF), testing. This conventional measurement approach typically ETH Zürich, Zürich, Switzerland 123 The International Journal of Advanced Manufacturing Technology has a drawback that the measurements need to be repeated industrial applications. It also allows the spindle to travel for every tool-holder combination used on a shop floor which through the workspace to capture potential position depen- makes the number of required experiments grow exponen- dencies of the machine tool dynamics. In the end, this device tially as the number of machines, tool holders, and tools eliminates the need for hammer impacts and can be used at increases. the end of the assembly line in machine tool manufacturing An alternative and commonly used path is to combine companies to characterize the machine tool dynamics and the experimentally measured dynamics of the machine tool provide their customers with this pivotal information. substructure with the analytically modeled dynamics of the The remainder of this paper is structured as follows: in tool-holder combination through receptance coupling sub- Section 2, two prototype designs of the spindle receptance structure analysis (RCSA) [12]. This is a promising approach identification device are described along with suitable recon- since analytical tool-holder models using Timoshenko beam struction methods. Section 3 presents different methods for elements accurately model the dynamics of tool-holder acquiring dynamics models of individual devices. Section 4 assemblies [13]. investigates the validation of reconstructed spindle dynam- Researchers have explored various methods for identify- ics versus other methods from the literature, dynamics at the ing machine dynamics. One approach involves analyzing tooltip, and prediction of stability boundaries. Finally, the cutting forces as a source of excitation and using result- paper is concluded in Section 5. ing vibrations to estimate the underlying dynamics [14, 15]. Alternatively, some researchers have relied on impact test- ing with a dummy cylindrical object mounted to the spindle 2 Methodology interface. This approach, known as inverse RCSA, allows for the mathematical subtraction of the dummy cylinder to This section outlines the underlying operational concept of obtain the dynamics [2, 16, 17]. The approach alleviates the the proposed automatic devices. Section 2.2 introduces the challenges of exciting rotational DOFs and measuring their substructure coupling concept, shedding light on the math- responses directly. Even in sophisticated studies that com- ematical feasibility of automatic reconstruction methods. bine machine learning with mathematical models [18, 19], it Building upon these foundations, it delves into the design is often assumed that the machine tool’s dynamics are known concept and the methods for reconstructing the spindle’s from the inverse RCSA method. However, it is important to dynamics in the two distinct layouts of the device, which note that this approach can be demanding in terms of conduct- are detailed in Sections 2.2.1 and 2.2.2. ing a large number of impact tests, especially if capturing the position-dependent machine dynamics across different posi- tions in the workspace is the goal [20] or when dealing with 2.1 Multi-component structural coupling concept numerous machine tools in real manufacturing companies. This paper presents an industry-friendly and accurate The presented approach consists of a measurement device identification method for the experimental measurement of with integrated piezoelectric actuators and accelerometers machine tool dynamics to automate the characterization of which is mounted to a machine tool spindle through a stan- the machine tool dynamics at the spindle interface. dard tool-holder interface (e.g., HSK A63). The embedded The proposed measurement devices are designed to be actuators excite the device and the connected machine tool easily mountable to a machine tool’s spindle interface anal- structure with the embedded sensors recording the response ogous to a tool holder. Multiple piezoelectric actuators are of the coupled structure. Finally, the dynamics of the machine embedded into the device structure that enable the excita- tool without the device are retrieved by mathematically tion of the spindle structure by applying alternating voltage, decoupling the known device dynamics. Figure 1 illustrates replicating a similar concept as an inertial shaker. Simultane- the conceptual coupling of a generic non-rigid structure rep- ously, integrated accelerometer sensors monitor the system resenting the device, denoted as substructure D, and the response at multiple locations leading to the collection of structure of machine tool and spindle, denoted as substruc- sufficient information for the estimation of translational and ture S. The resulting assembly of the machine tool structure rotational dynamics of the machine tool at its spindle flange. and device is denoted as structure A. Compared to other existing automated hammer designs [21] Capital variables denote frequency domain quantities that need external support and precise adjustments to align whose dependency on frequency s = j ω is omitted from the accurately with the intended direction of the excitation force, notation for simplicity in the remainder of this document. this design incorporates actuators and essential sensors into a Furthermore, variables in bold denote non-scalar quantities, singular unit, allowing for easy installation into the machine i.e., vectors or matrices. tool interface, thereby enhancing its practical feasibility for 123 The International Journal of Advanced Manufacturing Technology Furthermore, let H be the dynamics of the rigidly cou- A,ii pled system A, as illustrated in Fig. 1, such that = X = H F (3) A,i A,ii A,i The rigid coupling of substructures D and S is enforced by the compatibility and equilibrium conditions of X = X D,c S,c and F + F = 0, respectively. As a result, H can be D,c S,c A,ii Fig. 1 Generic structural coupling scheme of the measurement device expressed in terms of its substructure receptances through and spindle. The receptance coupling allows relating the dynamics of linear receptance coupling [22]: the assembly to the dynamics of its components −1 H = H − H ( H + H ) H (4) A,ii D,ii D,ic D,cc S,cc D,ci Let H denote the spindle receptance matrix which S,cc The device dynamics H can be obtained through a cali- relates frequency domain forces F to frequency domain S,c brated model of the device’s structure. The dynamics of the displacements X at the spindle interface: S,c coupled structure A is computed from the experimental mea- surements by the device when it is mounted on the machine. It X = H F (1) S,c S,cc S,c thus remains to estimate the spindle dynamics from the given data. Depending on the design of the measurement device, The subscript c indicates that the corresponding degrees of different reconstruction methods can be employed. freedom (DOFs) to the forces and displacements partici- pate in the coupling as will become more obvious once 2.2 Device layouts and reconstruction algorithms the device receptance has been presented. Axial and tor- sional dynamics of the spindle are usually neglected in In the presented approach, the excitation forces applied to the the context of chatter prediction in milling operations and assembled spindle-device structure (i.e., the components of the spindle receptance matrix remains confined to bending F ) are generated by embedded axial piezoelectric actu- A,i dynamics in a four-by-four matrix relating the force vec- ators, and the assembly’s responses are measured using tor F =[F , F , M , M ] to the displacement vector S,c x y x y accelerometers installed at multiple locations on the device. X =[X , Y ,θ ,θ ] . In this work, axis cross-coupling S,c x y As the actuators are embedded into the device, the effect components of the spindle dynamics are neglected, as they of a single actuator is a pair of external forces of equal mag- are roughly an order of magnitude smaller on common nitude and opposite directions at its supports. The quantities machine tool structures compared to in-axis components. that can be measured are the displacement vector X and A,i This simplifies the formulations to uni-directional cases, i.e., the vector of actuator input voltages V . Consequently, the A,i T T X =[X,θ ] and F =[F , M ] (and analogous for S,c x S,c x x measurable frequency response function (FRF) matrix H the Y direction). The presented equations hold also for the corresponds to X = H V = H T V where T A,i m A,i A,ii in A,i in more general case including axis cross-coupling terms. maps n independent actuator voltages to 2n external force Let H denote the device receptance matrix. The relevant inputs. The derivation of T for a single piezoelectric axial in DOFs of the device are divided into two groups: a group of actuator is further explained in Section 3, e.g., in Eq. 15.The DOFs that will be coupled with the spindle (subscript c) and measurements of the device then provide the following FRF a group of DOFs which will remain independent (subscript matrix: i). The uncoupled dynamic response of substructure D can then be described by the following: H = H T = H T m A,ii in D,ii in −1 − H ( H + H ) H T (5) D,ic D,cc S,cc D,ci in X F H H D,c D,c D,cc D,ci = H , where H = D D X F H H D,i D,i D,ic D,ii The goal now is to reconstruct H based on the known S,cc (2) dynamics of the assembly H and the device H . Equation 5 m D can be evaluated for any device, that fits the description of Here, X and X are vectors of frequency domain Eq. 2, as long as ( H + H ) is invertible, regardless D,c D,i D,cc S,cc displacements at coupled and independent coordinates, of the number of excitation inputs and sensors or their loca- respectively. Analogously, F and F are vectors of fre- tions. However, solving Eq. 5 in closed-form for H is D,c D,i S,cc quency domain forces at coupled and independent DOFs, only possible under certain conditions which are discussed in respectively. The detailed derivation of the device model is more detail in Section 2.2.2. A necessary condition is that the presented in Section 3. device offers equally many independent excitation inputs and 123 The International Journal of Advanced Manufacturing Technology sensors as components in F and X , respectively. Thus, Based on this spindle receptance model and the calibrated S,c S,c if the spindle receptance is considered to be a two-by-two dynamics of the device (see Section 3.1), Eq. 5 is evalu- receptance matrix, the device needs to offer two independent ated, and the predicted response H is compared to actual excitation inputs and two independent response sensors. measurements H at N frequency points ω through the fol- m i Based on this, two device prototypes are developed that lowing minimization loss function: are described in the following two subsections: The first one (Section 2.2.1) is simple and compact but does not offer suf- L ( H , H (θ )) = log (| H (ω )|)−log (| H (ω ,θ)|) m p m i p i 10 10 ficiently many excitation inputs for a closed-form spindle i =0 receptance reconstruction. Instead, a spindle reconstruction method based on a modal model of the spindle and gradient- +γ H (ω ) − H (ω ,θ) 1 m i p i based optimization is proposed. In the remainder of the i =0 document, this device is referred to as “one-stage device.” m +γ (max(0, −(ω − ω )) The other prototype (Section 2.2.2) is less compact but allows 2 k min k=0 to use a closed-form spindle reconstruction approach. The + max(0,ω − ω )) (7) k max term “two-stage device” shall refer to this device in the fol- lowing. with | H | and H representing the element-wise magnitude and phase of complex-valued matrix elements, respectively. 2.2.1 One-stage device and optimization-based spindle • also stands for Euclidean norm of matrices. In other receptance reconstruction words, the first two terms are the mean squared errors of the log magnitude and phase of the predicted and measured FRFs This prototype device is made from stainless steel and has the spindle interface of HSK A63 and the geometry shown in Fig. 2. It features a relatively compliant section in the middle, around which four axial piezoelectric actuators are arranged. Two opposing piezoelectric actuators are driven with voltages of equal magnitude and opposite polarity, hereby exciting bending modes of the structure. The device thus offers one independent excitation input for the X and one for the Y direction. To capture the dynamic response, six accelerometers—three in X and three in Y direction—are attached to the device. A modal model according to [23] (Eq. 3.74 in this refer- ence) is chosen to represent the spindle receptance matrix, as provided by the following: Rigid Coupling u u H (s) = (6) s + 2ζ ω s + ω k n,k n,k k=1 It assumes the linear behavior of the structure and inherently enforces the receptance matrix to be symmetric. Potential nonlinearities in the spindle receptance could be captured by making these modal parameters dependent on, e.g., static load or spindle speed, as is done by [24]. The number of Fig. 2 Structural arrangement of the intended machine tool with the modes m is a hyperparameter that needs to be determined in one-stage device. Highlighted piezoelectric actuators in gray are excited advance while the eigenfrequencies ω , damping ratios ζ , n,k k with opposite polarization to shake bending dynamics. Three indicated and mass-normalized mode shapes u are the optimization k accelerometers are meant to measure the lateral response (X direction) of the machine tool-device assembly parameters. Substructure D1 Substructure S (One-stage device) (Machine tool + Spindle) The International Journal of Advanced Manufacturing Technology Table 1 Overview of model Model parameter Reparametrization Initialization parameters, reparametrization, and initialization p k 1,k ω ω = e p = log(ω + (ω − ω )) n k 1,k min max min 2,k ζ ζ = e p = 0.01 k k 2,k ( p + jp ) 3,k,i 4,k,i u u = e p = p = 0.0 k,i k,i 3,k,i 4,k,i ω and ω are the bounds of the considered frequency range. m is a hyperparameter that sets the number min max of modes averaged over frequencies ω . The third term acts as a con- straint and encourages the eigenfrequencies to stay within the meaningful frequency range of (ω ,ω ). θ is the min max collection of optimization parameters, and γ and γ are 1 2 hyperparameters to balance the loss terms. For easier optimization, the model parameters are repara- metrized using the natural logarithm (for eigenfrequencies and damping ratios). This not only enforces them to remain positive but also allows easier exploration, i.e., to find the suitable order of magnitude more quickly. Similarly, the ele- ments of the mass-normalized mode shape u are expressed Rigid Coupling in terms of log magnitude and phase. An overview of optimization parameters and their reparametrizations and ini- tializations is given in Table 1. 1 1 PyTorch [25] is used for the implementation of the spindle model, device model, and loss function which allows using its automatic differentiation functionality to obtain gradients of the loss function. The Adam optimizer [26]isemployed to minimize the loss function using batch gradient descent. 2.2.2 Two-stage device and closed-form spindle receptance reconstruction X-direction Y-direction This device is made from stainless steel and has a spin- dle interface of HSK A63 similar to the one-stage device. Fig. 3 Structural arrangement of the intended machine tool with the However, at the cost of being less compact, the two-stage two-stage device. Highlighted piezoelectric actuators in gray and light device features a second stage of piezoelectric actuators red provide two independent excitations to machine tool-device assem- (Fig. 3). Again, two opposing piezoelectric actuators are dri- bly while the indicated accelerometers record the resulting lateral response. The indicated jamming bolts in the Y direction are used to ven with voltages of equal magnitude and opposite polarity. disturb symmetry in device dynamics in X and Y directions The two stages of piezoelectric actuators offer two inde- pendent excitation inputs per X and Y directions. Con- sequently, a closed-form spindle dynamics reconstruction method can be used, which is described next. Solving Eq. 5 to be invertible. The inverses of the latter two terms are for H yields the following: required in solving for H . In words, the invertibility S,cc S,cc of ( H T ) implies that the actuators embedded in the D,ci in −1 H = H T ( H T − H ) H − H (8) device need to be able to cause displacements in all DOFs S,cc D,ci in D,ii in m D,ic D,cc of X = X . Thus, if X =[X,θ ] , then two D,c S,c S,c x As H is expected to be symmetric, the off-diagonal com- actuators are needed that cause two linearly independent S,cc ponents are averaged, i.e., H = 0.5( H + H ). displacement vectors at the coupling point. Similarly, the S,cc,sym S,cc S,cc Equation 8 requires ( H T − H ), ( H T ), and H invertibility of H requires that any force vector F D,ii in m D,ci in D,ic D,ic D,c Substructure D2 Substructure S (Two-stage device) (Machine tool + Spindle) The International Journal of Advanced Manufacturing Technology Fig. 4 Translational and rotational components of the spindle dynamics second case, the Y -axis of the device was aligned with the X-axis of the of a GFMS Mikron HPM800U 5-axis milling machine in X direction machine. The two predictions are corrupted around different frequen- as estimated by the two-stage device in two different orientations and cies (approximately 1500 Hz and 1700 Hz), because the dynamics of a reference obtained as suggested in [27]. In one case, the X-axis of the two-stage device is different in X and Y directions the device was aligned with the X-axis of the machine while in the applied at the coupling point causes a unique displacement the X-axis of the machine, and once with the Y -axis of the vector X that can be captured by the embedded sensors. device aligned with the X-axis of the machine (i.e., a 90 D,i Hence, if F =− F , two sensors must be placed in rotation of the device between measurements). The same pro- S,c D,c different locations. Finally, the invertibility requirement of cedure should be carried out for measuring the Y -axis of the ( H T − H ) means that the measurements obtained machine. This essentially yields two estimates of the spin- D,ii in m from the device when mounted to the spindle should be dif- dle receptance from two different device dynamics (denoted ferent from measurements obtained from substructure D (the H and H ) per direction. Consequently, the S,cc1,sym S,cc2,sym device without HSK section) alone. Note that the above con- spindle receptance estimates obtained using Eq. 8 are cor- ditions must hold on a frequency-by-frequency basis in the rupted around different frequencies, as can be seen in Fig. 4. frequency range of interest. The FRF components of the two estimates are then com- The spindle receptance estimates resulting from Eq. 8 are bined by computing their weighted arithmetic mean. The corrupted by inevitable model mismatch, which becomes weights for measurements 1 and 2 and are defined as particularly pronounced around eigenfrequencies of the device where the term that needs to be inverted may become |σ (ω)| 1,ij r (ω) = 1 − ill-conditioned. Reconstruction errors can be reduced if the 1,ij |σ (ω)|+|σ (ω)| 1,ij 2,ij eigenfrequencies of the device are chosen to be outside the |σ (ω)| 2,ij frequency range of interest or if the device model can be r (ω) = 1 − (9) 2,ij |σ (ω)|+|σ (ω)| 1,ij 2,ij determined with extremely high accuracy. If neither of these requirements can be satisfied, a remaining solution is to repeat the spindle receptance measurement with two different where σ (ω) is the sliding window standard deviation (con- k,ij device dynamics, i.e., different in terms of their eigenfre- sidering a window length of 2 P + 1 samples) of the FRF quencies. This approach is chosen to obtain accurate spindle component ij of measurement k at frequency point ω com- receptance estimates with the available prototype. The device puted as follows: dynamics in the Y direction are altered by jamming a pair of bolts in the lower piezo stage, as shown in Fig. 3. The measurement of the spindle receptance should be then ¯ 2 σ (ω ) = (H (ω ) − H ) (10) k,ij l k,ij l+ p k,ij performed once with the X-axis of the device aligned with 2 P + 1 p=− P 123 The International Journal of Advanced Manufacturing Technology , coupled machine tool-device dynamic behavior according to Eq. 4. Due to the design of the prototype devices described in Sections 2.2.1 and 2.2.2, its lateral dynamics in X and Y directions are expected to be independent of each other (i.e., no axis cross-coupling). The models of the device 2 2 dynamics presented in the following are thus restricted to uni-directional translational and rotational deformations. The , , HSK-A63 interface is not part of the device models, as it is , , 1 1 1 1 considered to be part of the machine tool. In Section 3.1, it is suggested that utilizing a simplified , , 1 1 1 1 Timoshenko beam model could yield an adequately pre- , , cise model for the one-stage device and optimization-based spindle receptance reconstruction approach. The two-stage 3 3 device, on the other hand, involves more components and complexities, and hence, developing a physics-based device model through modeling and calibration could be an exten- sive task. To provide an alternative solution, Section 3.2 presents an experimental approach based on inverse RCSA 4 4 , for obtaining device models. 5 5 3.1 Physics-based modeling and calibration The receptance of the one-stage prototype device is modeled through sequential receptance coupling of cylinders under axial or lateral load, as detailed in Section 3.1.2.The Timo- shenko beam model is used to compute the receptance of a single cylinder subject to lateral forces and bending moments at its ends. For the axial receptance of a uniform rod, the ana- lytical solutions provided in [28] are used. The cylindrical piezoelectric axial actuators embedded in the device are mod- Fig. 5 Dynamics model of the one-stage device: Cylinders (drawn as eled as mere elastic structures. The converse piezoelectric rectangles) subject to lateral or axial load are assembled through rigid effect is modeled by a pair of external forces as described in and elastic receptance coupling. External force inputs (blue) model the Section 3.1.1, following the approach taken in [29]. piezoelectric effect of the applied voltage. Displacement outputs are denoted in red 3.1.1 Piezoelectric actuator Before computing the weighted average, the two mea- surements are smoothed using moving median filtering on Let the axial receptance of a linear elastic rod be given magnitude and phase. Finally, the resulting spindle recep- through the following receptance matrix: tance is computed as H = r H + S,cc,av,ij 1,ij S,cc1,sym,ij r H . The FRFs of the device are obtained as 2,ij S,cc2,sym,ij Z H H F 1 11 12 z1 = (11) described in Section 3.2. Z H H F 2 21 22 z2 The elongation of a piezoelectric material due to the volt- 3 Bending dynamics of the measurement age V applied across its end faces can be described by devices This section is devoted to obtaining the required dynamics z1 Z − Z = H − H H − H + d V (12) 2 1 21 11 22 12 33 of the two device configurations as they are involved in the z2 123 The International Journal of Advanced Manufacturing Technology where the force-induced portion of the deformation is based 3.1.3 Model calibration on the axial receptance according to Eq. 11 and d is the piezoelectric coefficient. The voltage-induced portion of the To capture the dynamics of the device alone (i.e., not attached elongation is now replaced by a pair of external forces of to any machine tool), it is hung from an elastic string to equal magnitude F and opposite directions: simulate free-free boundary conditions. FRF measurements ext are then obtained from the actuators and sensors embedded in the device as well as from tap testing on two points on the F + F z1 ext Z − Z = H − H H − H (13) 2 1 21 11 22 12 tool-holder part. F − F z2 ext A Timoshenko beam-based model of the tool-holder taper is added to the device model through receptance coupling, as Comparing the above two equations yields an expression it is considered to be part of the machine tool and is thus not for the magnitude of the external force in terms of applied included in the device model. The device model is then cal- voltage and receptance of the piezoelectric material: ibrated by tuning model parameters using genetic algorithm optimization, such that the obtained measurements match d V F = (14) ext the FRFs predicted by the device model. All of the afore- H + H − H − H 21 12 11 22 , c ) as well as the outer mentioned contact parameters (k i i diameter and Young’s modulus of the center bolt are consid- The mapping from actuator inputs (a single excitation volt- ered optimization parameters. The properties of the center age in this case) to external forces reads as follows: bolt are tuned to take the effect of the two/four piezo stacks into account which is not considered in this planar model. F 1 33 z1,ext = T V = V in F −1 H + H − H − H z2,ext 21 12 11 22 3.2 Experimental modeling through inverse RCSA (15) This approach aims to obtain all required components of 3.1.2 Assembled model through coupling substructures the device FRFs through measurements. The obstacle here is applying rotational moments at the coupling point with An overview of the dynamics models of the one-stage device the spindle, which is not practical in reality. To circum- is given in Fig. 5. The cylinders are drawn as rectangles; vent this issue, the top part of the device is modeled using their center axes as dash-dotted lines. Solid lines denote rigid a Timoshenko beam model and is considered known. The couplings whereas linear and rotational elastic couplings are FRF measurements are taken from the device in free-free indicated by the respective pictograms. The latter has been condition using the embedded actuators and sensors on the introduced between the piezoelectric elements and the neigh- unknown part of the device, as well as through impacting boring components (k , c ). They are supposed to model the 1 1 and measuring on the known part of the device (Fig. 7). The effect of the 0.1 mm thin copper electrodes and a 0.07-mm FRFs of the unknown part of the device can then be obtained thin plastic insulation sheet which are found in these locations using inverse RCSA. Finally, part of the subtracted structure in the physical prototype. Axial forces and displacements at is added again to get the model of the device up to the desired the ends of the two piezo stacks are mapped to moments and coupling location. A graphical overview of the method is rotational displacements through linear mapping. given in Fig. 6. Elastic couplings are further introduced between this For obtaining the measurements, the device was again assembly of piezo stacks and the surrounding frame (k , c 2 2 hung from an elastic string to simulate free-free boundary and k , c ). They should take into account any local defor- conditions, as shown in Fig. 7. 3 3 mations and contact stiffnesses. Finally, linear and rotational spring-damper systems are introduced where the physical device is bolted together (k , c , k , c ). The free inputs and 4 4 5 5 4 Experimental validations outputs of the device model are indicated in blue and red, respectively. For the one-stage device, they correspond to The two versions of the measurement device are used to esti- F and X from Eq. 2 as follows: D,i D,i mate the spindle receptance of a GFMS Mikron HPM800U 5-axis milling machine in the setup shown in Fig. 8.An exponential chirp excitation ranging from approximately 80 F = F F F F (16) D,i D,i0 D,i1 D,i2 D,i3 Hz to 6 kHz with a voltage amplitude slightly below 100 X = X X X (17) D,i D,i0 D,i1 D,i2 V is used for both devices. The excitation and response 123 The International Journal of Advanced Manufacturing Technology F X Si,0 Si,0 known Di,0 known X = X Di,0 Si,1 Si,1 Di,0 Inverse RCSA known Di,1 Di,1 Di,2 Fig. 6 Device dynamics identification method based on inverse RCSA, where the FRFs of unknown substructures are determined by subtracting theoretical models of known parts from the measurements of the intact device signals are sampled at 51.2 kHz using a National Instru- the resulting tooltip dynamics using RCSA to impact test- ments data acquisition card type 9234. PCB Piezotronics ing measurements. Seven different tool-holder combinations accelerometers of type 352C22 are employed to capture the response. The measurements are repeated 100 times to aver- age out measurement noise. The number of modes of the modal spindle model used with the one-stage device and the optimization-based spindle receptance reconstruction is chosen to be m = 10 as a hyperparameter. The resulting modal parameters can be found in Appendix A.For thetwo- stage device, measurements are conducted as multiple SIMO (single input multi output) experiments, using one excitation input at a time and setting the other voltage input to zero by short-circuiting the electrodes of the piezoelectric actuators. 4.1 Validations of estimated spindle dynamics versus manual impact testing The estimated spindle dynamics by the two device layouts are compared against reference values in Fig. 9.The refer- ence receptance is obtained through an inverse receptance coupling approach as suggested by Namazi et al. [27]. In this method, a cylindrical dummy holder is mounted to the machine, and FRFs are measured at certain locations through impact testing. This indirect measurement method is proposed since the excitation of rotational dynamics by a hammer is not straightforward. In the following, the mea- surements of the machine’s dynamics at the spindle flange are compared against each other. 4.2 Validations through tooltip dynamics As mentioned earlier, the structural dynamics at the tooltip determine the boundaries of stable process conditions. Hence, it is valuable to evaluate the validity of the reconstructed spindle dynamics from the proposed devices by comparing Fig. 7 The two-stage device during structural identification. To simu- late free-free boundary conditions, it is hung from an elastic string 123 The International Journal of Advanced Manufacturing Technology Fig. 8 The prototype devices mounted to a GFMS Mikron HPM800U 5-axis machining center. Left: one-stage device. Right: two-stage device are selected for this validation purpose. The details of these ibility and equilibrium conditions respectively, the coupled combinations are provided in Table 2. The dynamics of the dynamics of the system at the tooltip can be computed as tooling systems are obtained through a finite element mod- follows: eling approach based on Timoshenko beam elements with −1 considering a non-uniformly distributed contact flexibility H = H − H ( H + H ) H (18) ST ,11 T ,11 T ,12 S,33 T ,22 T ,21 at the tool-holder interface. More details and corresponding values of the contact parameters can be found in [13]. Further details of substructure coupling through the RCSA Considering the substructuring scenario presented in method can be found in publications of Schmitz et al. [30, Fig. 10 with X = X and F + F = 0 as compat- T ,2 S,3 T ,2 S,3 31]. Fig. 9 Translational and rotational components of the spindle dynamics of a GFMS Mikron HPM800U 5-axis milling machine in X and Y directions as estimated by the one-stage device, the two-stage device, and the reference obtained through iRCSA 123 The International Journal of Advanced Manufacturing Technology Table 2 Tooling systems (S.L. # Holder Tool S.L. (mm) Picture indicates the stickout length of the cutting tool) 1 Zürn HSK-A63 63.11.20.2 Voha 12032456120 70.50 Interface: Collet ER32 4 flutes 2 REGO-FIX PG25x100H Diameter: 12.0 mm 65.90 Interface: powRgrip PG25 Length: 110.5 mm 3 REGO-FIX PG25x100H 60.90 Interface: powRgrip PG25 4 Schunk 208123 50161717 62.60 Interface: Thermal shrink fit 5 Zürn HSK-A63 11.16.23 Voha 2002805 Interface: Collet ER25 4 flutes 60.07 Diameter: 16.0 mm Length: 93.3 mm 6 Long insert tool 5 cutters − Diameter: 40.0 mm 7 Short insert tool 5 cutters − Diameter: 50.0 mm Figure 11 shows tooltip receptance measurements for the 4.3 Validations of predicted stability boundaries tool-holder combinations described in Table 2 along with the against experimental observations predictions using on the spindle receptance measurements from the one-stage device, the two-stage device, and the ref- Experimental observations of stability states and correspond- erence dynamics through iRCSA. ing chatter frequencies are used to verify the model-based prediction of SLDs through the zero-order approximation method by Altintas and Budak [3]. The required tooltip dynamics are taken from the RCSA coupling presented in the preceding section. The validation cuts are collected through experimental cutting tests for the first two tool-holder combi- nations from Table 2 on a block of Al6082 aluminum. Table 3 summarizes the corresponding process parameters. In the prediction of stability charts, the experimentally calibrated values of 902 MPa and 243 MPa are assumed for the tangen- tial and radial cutting force coefficient, respectively. An acceptable overall agreement between the three pre- dictions and the validation data can be observed for the first validation case in Fig. 12a. In the second case (Fig. 12b), the predictions based on the reference spindle receptance Rigid Coupling and the estimate from the two-stage device are in agree- ment, but the predicted critical depth of cut based on the spindle receptance estimate from the one-stage device is slightly higher. This is due to an inaccuracy in the magni- tude of the corresponding tooltip receptance in Y direction (Fig. 11b), which is underestimated for frequencies around the dominant mode (2100–2300 Hz). It is worth noting that the SLDs generated using the measured tooltip FRFs show slight deviations from those produced by the predicted FRFs. Fig. 10 Coupling scenario of the machine tool structure and tooling However, this deviation is not necessarily positive in terms system. The model for tooling systems is obtained from the method of improved prediction accuracy of the stability borders ver- presented in [13] Substructure T Substructure S (Tool-holder assembly) (Machine tool + Spindle) The International Journal of Advanced Manufacturing Technology Fig. 11 Tooltip receptance predictions and measurement for tool- using the method presented in [27]. These spindle receptances are holder combinations #1 to #7 in Table 2 correspond to a to g, rigidly coupled with Timoshenko-beam-based tooling models to obtain respectively. Predictions are based on spindle receptance estimations tooltip predictions from the one-stage device, two-stage device, and a reference obtained 123 The International Journal of Advanced Manufacturing Technology Table 3 Process parameters Tooling Feed Milling Radial depth Feed rate used for the validation cuts with Case system direction Strategy of cut (mm) (mm/tooth) two tool-holder combinations (according to Table 2)ona 11 Y + Slotting 12.0 0.05 block of Al6082 aluminum 22 X+ Up-milling 8.4 0.05 sus the validation cuts. This raises the possibility that the more, the spindle receptance reconstruction method could direct measurement of tooltip FRFs using an impact hammer be expanded to capture axis cross-couplings, which involves and accelerometer may be compromised. The imprecision the response of the spindle in the X direction to the excitation may stem from misalignment in the direction and location of in the Y direction and vice versa. hammer impacts or the accelerometer, particularly given the fluted geometry of the tooltip. 5 Conclusion This study introduced two prototypes of measurement devices and their identification methods to automatically identify machine tool dynamics at the spindle interface. The two devices differ in the number of independent piezoelec- tric actuators and embedded acceleration sensors, as well as in their compactness. Both approaches produced satisfactory results. The estimation of spindle dynamics using the one-stage device provides a compact measurement system with accept- able measurement accuracy. In the underlying optimization- based reconstruction method, like any optimization problem, attention must be given to the convergence to local min- (a) imums, such as when translational and rotational FRFs compensate for each other. Further inclusion of physical con- straints or prior knowledge can be helpful for the convergence of the optimization to global solutions. The two-stage device and the closed-form reconstruc- tion method for spindle receptance estimation led to faster, unique, and repeatable results. However, the device com- pactness is compromised due to the additional stage of piezoelectric actuators. Future work could explore a more sophisticated arrangement of piezoelectric actuators, such as the Stewart-platform arrangement [32], which offers independent excitation inputs while maintaining a compact design. Improving the accuracy of the device model can further enhance the precision of the reconstructed spindle dynamics, which encourages arrangements that can be more reliably modeled in the design stage. The two proposed measurement devices enabled the iden- tification of machine tool receptance with high repeatability and minimal human effort, as only mounting the device on (b) the machine tool is required. Additionally, the approach is Fig. 12 Predicted stability boundaries and experimental chatter obser- suitable for measuring the machine tool dynamics in various vations for tool-holder combination #1 in a and #2 in b from Table 2 locations in the working space since the device can be easily using process parameters according to Table 3 on a block of Al6082 moved around without requiring external support. Further- aluminum 123 The International Journal of Advanced Manufacturing Technology Appendix A: Modal parameters of estimated spindle receptances Table 4 Modal parameters of k ω [rad/s] ζ [−] u [ s/kg] u [rad s/kgm ] n,k k k,0 k,1 the spindle receptance in X direction as estimated by the 1 1924.67 0.003 0.01134-0.00145j −0.0271+0.0310j one-stage device and 2 4518.64 0.163 0.17878+0.025827j 1.9041+0.89598j optimization-based identification method 3 8562.86 0.028 0.23317−0.079048j 0.80471−0.085383j 4 9125.85 0.461 0.14483−0.72793j 2.7327−9.7517j 5 10,890.80 0.165 0.67985−0.64180j 2.2236−6.5233j 6 12,421.76 0.131 1.0155−0.10138j 5.3805−4.5614j 7 14,057.43 0.155 1.0305+0.35473j 9.6915−3.0552j 8 21,090.84 0.231 0.96087+0.61073j 0.31771+0.63735j 9 22,391.05 0.548 1.3746+1.3535j 36.942+10.874j 10 22,808.70 0.224 −0.849+0.97204j −3.3365+18.436j Table 5 Modal parameters of √ k ω [rad/s] ζ [−] u [ s/kg] u [rad s/kgm ] n,k k k,0 k,1 the spindle receptance in Y direction as estimated by the 1 2239.41 0.113 0.03061−0.044494j 0.0235+0.00492j one-stage device and 2 4584.90 0.111 0.1209−0.03875j 1.8830−0.32124j optimization-based identification method 3 8107.34 0.141 0.29544+0.0776j −2.9079−3.5816j 4 9437.61 0.031 0.1990−0.09720j 0.8738−1.8657j 5 10,271.24 0.109 0.28643+0.1873j 0.9037−2.8547j 6 15,633.82 0.251 1.2377−0.46629j 8.3055−0.17174j 7 17,837.63 0.079 0.43768+0.15219j 4.8724−5.0213j 8 20,598.82 0.162 0.7702+0.12895j 0.11743−13.038j 9 21,461.37 0.441 0.75704+1.3509j 36.08+15.436j 10 23,434.09 0.023 0.2007+0.042563j 0.20581+0.29781j 123 The International Journal of Advanced Manufacturing Technology Author Contributions VOAA: conceptualization, methodology, super- 14. Akbari VOA, Mohammadi Y, Kuffa M, Wegener K (2023) Identifi- vision, final draft editions. CS: methodology, analysis, writing original cation of in-process machine tool dynamics using forced vibrations draft, experimental setup, data curation. MK and KW: supervision, in milling process. Int J Mech Sci 239:107887 review, and funding acquisition. 15. Liu X, Cheng K (2005) Modelling the machining dynamics of peripheral milling. Int J Mach Tools Manuf 45(11):1301–1320 Funding Open access funding provided by Swiss Federal Institute of 16. Montevecchi F, Grossi N, Scippa A, Campatelli G (2017) Two- points-based receptance coupling method for tool-tip dynamics Technology Zurich. This research was financially supported by Inno- prediction. Mach Sci Technol 21(1):136–156 suisse, the Swiss Innovation Agency (grant 32334.1 IP-ICT). 17. Namazi M, Altintas Y, Abe T, Rajapakse N (2007) Modeling and identification of tool holder-spindle interface dynamics. Int J Mach Declarations Tools Manuf 47(9):1333–1341 18. Akbari VOA, Kuffa M, Wegener K (2023) Physics-informed Bayesian machine learning for probabilistic inference and refine- Conflict of interest The authors declare no competing interests. ment of milling stability predictions. CIRP J Manuf Sci Technol 45:225–239 Open Access This article is licensed under a Creative Commons 19. Wegener K, Weikert S, Mayr J, Maier M, Ali Akbari VO, Postel M Attribution 4.0 International License, which permits use, sharing, adap- (2021) Operator integrated-concept for manufacturing intelligence. tation, distribution and reproduction in any medium or format, as J Mach Eng 21 long as you give appropriate credit to the original author(s) and the 20. Postel M, Bugdayci NB, Monnin J, Kuster F, Wegener K (2018) source, provide a link to the Creative Commons licence, and indi- Improved stability predictions in milling through more realistic cate if changes were made. The images or other third party material load conditions. Proc CIRP 77:102–105 in this article are included in the article’s Creative Commons licence, 21. Postel M, Candia N, Bugdayci B, Kuster F, Wegener K (2019) unless indicated otherwise in a credit line to the material. If material Development and application of an automated impulse hammer is not included in the article’s Creative Commons licence and your for improved analysis of five-axis CNC machine dynamics and intended use is not permitted by statutory regulation or exceeds the enhanced stability chart prediction. Int J Mechatron Manuf Syst permitted use, you will need to obtain permission directly from the copy- 12(3–4):318–343 right holder. To view a copy of this licence, visit http://creativecomm 22. Ferreira J, Ewins D (1996) Nonlinear receptance coupling approach ons.org/licenses/by/4.0/. based on describing functions. In: Proceedings-SPIE the interna- tional society for optical engineering, pp. 1034–1040 23. 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Wu J, Song Y, Liu Z, Li G (2022) A modified similitude analy- 28. Schmitz TL, Smith KS (2021) Mechanical vibrations. Springer sis method for the electro-mechanical performances of a parallel International Publishing. https://doi.org/10.1007/978-3-030- manipulator to solve the control period mismatch problem. Sci 52344-2 China Technol Sci 65(3):541–552 29. Ostad Ali Akbari V, Ahmadi K (2021) Substructure analysis 8. Wu J, Yu G, Gao Y, Wang L (2018) Mechatronics modeling and of vibration-assisted drilling systems. Int J Adv Manuf Technol vibration analysis of a 2-DOF parallel manipulator in a 5-DOF 113:2833–2848. https://doi.org/10.1007/s00170-021-06777-1 hybrid machine tool. Mech Mach Theory 121:430–445 30. Schmitz TL, Smith KS (2001) Mechanical vibrations: modeling 9. Cheng K (2008) Machining dynamics: fundamentals, applications and measurement and practices. Springer Science & Business Media 31. Schmitz TL, Davies MA, Kennedy MD (2001) Tool point fre- 10. Ganguly V, Schmitz TL (2013) Spindle dynamics identification quency response prediction for high-speed machining by RCSA. J using particle swarm optimization. J Manuf Process 15(4):444– Manuf Sci Eng 123(4):700–707 451 32. Stewart D (1965) A platform with six degrees of freedom. Proc Inst Mech Eng 180(1):371–386 11. Cao Y, Altintas Y (2004) A general method for the modeling of spindle-bearing systems. J Mech Des 126(6):1089–1104 12. Schmitz TL, Donalson R (2000) Predicting high-speed machining dynamics by substructure analysis. CIRP Ann 49(1):303–308 Publisher’s Note Springer Nature remains neutral with regard to juris- 13. Ostad Ali Akbari V, Postel M, Kuffa M, Wegener K (2022) Improv- dictional claims in published maps and institutional affiliations. ing stability predictions in milling by incorporation of toolholder sound emissions. CIRP J Manuf Sci Technol 37:359–369 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The International Journal of Advanced Manufacturing Technology Springer Journals

Automated machine tool dynamics identification for predicting milling stability charts in industrial applications

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Abstract

As the machine tool dynamics at the tooltip is a crucial input for chatter prediction, obtaining these dynamics for industrial applications is neither feasible through experimental impact testing for numerous tool-holder-spindle combinations nor feasible through physics-based modeling of the entire machine tool due to their sophisticated complexities and calibrations. Hence, the often-chosen path is a mathematical coupling of experimentally measured machine tool dynamics to model-predicted tool-holder dynamics. This paper introduces a novel measurement device for the experimental characterization of machine tool dynamics. The device can be simply mounted to the spindle flange to automatically capture the corresponding dynamics at the machine tool side, eliminating the need for expertise and time-consuming setup efforts thus presenting a viable solution for industries. The effectiveness of this method is evaluated against conventional spindle receptance measurement attempts using impact tests. The obtained results are further validated in the prediction of tooltip dynamics and stability boundaries. Keywords Chatter stability · Machine tool dynamics · Piezoelectric actuators · Structural coupling · Inverse RCSA 1 Introduction Regardless of the stability models being formulated in frequency [2–4] or time domain [5, 6], the generation of Until today, regenerative chatter has been the main factor SLDs requires precise knowledge of cutting forces and the limiting the productivity of milling operations in manufac- dynamics at the tip of end mills, where the cutting forces turing industries. Despite the extensive research on this topic excite the machine tool structure. While the cutting forces since its first attention by Tlusty and Polacek [1] in 1963, a can be characterized by simplified coefficients [2], the com- solid solution implementable in an industrial environment is plexity of obtaining machine tool dynamics has remained the still lacking. major obstacle in industrial profits from SLD predictions. It While stability lobe diagrams (SLDs) provide a powerful is worth noting that the knowledge of system dynamics is solution to predict the boundary between stable and unstable not merely useful in SLD creation, but also essential for con- axial depth of cut as a function of spindle speed [2], they are trol applications, where the interaction between mechanical barely used in industries due to the significant measurement and mechatronic subsystems determines the performance of and modeling effort that is required for their creation. machine tool controllers [7, 8]. Further insights into applica- An existing SLD allows for a proper selection of depth tions of machine dynamics can be found in [9]. of cut and spindle speed leading to a significant productivity Modeling the entire machine tool structure including the increase while chatter and its negative consequences are pre- tool-holder assembly is challenging, due to missing informa- vented. However, a blind selection of process parameters may tion (such as the bearing locations) and the unknown damping lead to poor surface quality, significant tool wear, damage to characteristics of the numerous joints of the machine tool the machine tools, increased production cost, and excessive structure [10, 11]. Hence, a direct experimental measurement noise emissions. of the tooltip dynamics using impact hammers may seem an immediate option. While direct measurement of tooltip dynamics can be used for the generation of a precise SLD, B Vahid Ostad Ali Akbari it can be very time-consuming and prone to errors due to [email protected] inaccurate impact location and direction during manual tap Institute of Machine Tools and Manufacturing (IWF), testing. This conventional measurement approach typically ETH Zürich, Zürich, Switzerland 123 The International Journal of Advanced Manufacturing Technology has a drawback that the measurements need to be repeated industrial applications. It also allows the spindle to travel for every tool-holder combination used on a shop floor which through the workspace to capture potential position depen- makes the number of required experiments grow exponen- dencies of the machine tool dynamics. In the end, this device tially as the number of machines, tool holders, and tools eliminates the need for hammer impacts and can be used at increases. the end of the assembly line in machine tool manufacturing An alternative and commonly used path is to combine companies to characterize the machine tool dynamics and the experimentally measured dynamics of the machine tool provide their customers with this pivotal information. substructure with the analytically modeled dynamics of the The remainder of this paper is structured as follows: in tool-holder combination through receptance coupling sub- Section 2, two prototype designs of the spindle receptance structure analysis (RCSA) [12]. This is a promising approach identification device are described along with suitable recon- since analytical tool-holder models using Timoshenko beam struction methods. Section 3 presents different methods for elements accurately model the dynamics of tool-holder acquiring dynamics models of individual devices. Section 4 assemblies [13]. investigates the validation of reconstructed spindle dynam- Researchers have explored various methods for identify- ics versus other methods from the literature, dynamics at the ing machine dynamics. One approach involves analyzing tooltip, and prediction of stability boundaries. Finally, the cutting forces as a source of excitation and using result- paper is concluded in Section 5. ing vibrations to estimate the underlying dynamics [14, 15]. Alternatively, some researchers have relied on impact test- ing with a dummy cylindrical object mounted to the spindle 2 Methodology interface. This approach, known as inverse RCSA, allows for the mathematical subtraction of the dummy cylinder to This section outlines the underlying operational concept of obtain the dynamics [2, 16, 17]. The approach alleviates the the proposed automatic devices. Section 2.2 introduces the challenges of exciting rotational DOFs and measuring their substructure coupling concept, shedding light on the math- responses directly. Even in sophisticated studies that com- ematical feasibility of automatic reconstruction methods. bine machine learning with mathematical models [18, 19], it Building upon these foundations, it delves into the design is often assumed that the machine tool’s dynamics are known concept and the methods for reconstructing the spindle’s from the inverse RCSA method. However, it is important to dynamics in the two distinct layouts of the device, which note that this approach can be demanding in terms of conduct- are detailed in Sections 2.2.1 and 2.2.2. ing a large number of impact tests, especially if capturing the position-dependent machine dynamics across different posi- tions in the workspace is the goal [20] or when dealing with 2.1 Multi-component structural coupling concept numerous machine tools in real manufacturing companies. This paper presents an industry-friendly and accurate The presented approach consists of a measurement device identification method for the experimental measurement of with integrated piezoelectric actuators and accelerometers machine tool dynamics to automate the characterization of which is mounted to a machine tool spindle through a stan- the machine tool dynamics at the spindle interface. dard tool-holder interface (e.g., HSK A63). The embedded The proposed measurement devices are designed to be actuators excite the device and the connected machine tool easily mountable to a machine tool’s spindle interface anal- structure with the embedded sensors recording the response ogous to a tool holder. Multiple piezoelectric actuators are of the coupled structure. Finally, the dynamics of the machine embedded into the device structure that enable the excita- tool without the device are retrieved by mathematically tion of the spindle structure by applying alternating voltage, decoupling the known device dynamics. Figure 1 illustrates replicating a similar concept as an inertial shaker. Simultane- the conceptual coupling of a generic non-rigid structure rep- ously, integrated accelerometer sensors monitor the system resenting the device, denoted as substructure D, and the response at multiple locations leading to the collection of structure of machine tool and spindle, denoted as substruc- sufficient information for the estimation of translational and ture S. The resulting assembly of the machine tool structure rotational dynamics of the machine tool at its spindle flange. and device is denoted as structure A. Compared to other existing automated hammer designs [21] Capital variables denote frequency domain quantities that need external support and precise adjustments to align whose dependency on frequency s = j ω is omitted from the accurately with the intended direction of the excitation force, notation for simplicity in the remainder of this document. this design incorporates actuators and essential sensors into a Furthermore, variables in bold denote non-scalar quantities, singular unit, allowing for easy installation into the machine i.e., vectors or matrices. tool interface, thereby enhancing its practical feasibility for 123 The International Journal of Advanced Manufacturing Technology Furthermore, let H be the dynamics of the rigidly cou- A,ii pled system A, as illustrated in Fig. 1, such that = X = H F (3) A,i A,ii A,i The rigid coupling of substructures D and S is enforced by the compatibility and equilibrium conditions of X = X D,c S,c and F + F = 0, respectively. As a result, H can be D,c S,c A,ii Fig. 1 Generic structural coupling scheme of the measurement device expressed in terms of its substructure receptances through and spindle. The receptance coupling allows relating the dynamics of linear receptance coupling [22]: the assembly to the dynamics of its components −1 H = H − H ( H + H ) H (4) A,ii D,ii D,ic D,cc S,cc D,ci Let H denote the spindle receptance matrix which S,cc The device dynamics H can be obtained through a cali- relates frequency domain forces F to frequency domain S,c brated model of the device’s structure. The dynamics of the displacements X at the spindle interface: S,c coupled structure A is computed from the experimental mea- surements by the device when it is mounted on the machine. It X = H F (1) S,c S,cc S,c thus remains to estimate the spindle dynamics from the given data. Depending on the design of the measurement device, The subscript c indicates that the corresponding degrees of different reconstruction methods can be employed. freedom (DOFs) to the forces and displacements partici- pate in the coupling as will become more obvious once 2.2 Device layouts and reconstruction algorithms the device receptance has been presented. Axial and tor- sional dynamics of the spindle are usually neglected in In the presented approach, the excitation forces applied to the the context of chatter prediction in milling operations and assembled spindle-device structure (i.e., the components of the spindle receptance matrix remains confined to bending F ) are generated by embedded axial piezoelectric actu- A,i dynamics in a four-by-four matrix relating the force vec- ators, and the assembly’s responses are measured using tor F =[F , F , M , M ] to the displacement vector S,c x y x y accelerometers installed at multiple locations on the device. X =[X , Y ,θ ,θ ] . In this work, axis cross-coupling S,c x y As the actuators are embedded into the device, the effect components of the spindle dynamics are neglected, as they of a single actuator is a pair of external forces of equal mag- are roughly an order of magnitude smaller on common nitude and opposite directions at its supports. The quantities machine tool structures compared to in-axis components. that can be measured are the displacement vector X and A,i This simplifies the formulations to uni-directional cases, i.e., the vector of actuator input voltages V . Consequently, the A,i T T X =[X,θ ] and F =[F , M ] (and analogous for S,c x S,c x x measurable frequency response function (FRF) matrix H the Y direction). The presented equations hold also for the corresponds to X = H V = H T V where T A,i m A,i A,ii in A,i in more general case including axis cross-coupling terms. maps n independent actuator voltages to 2n external force Let H denote the device receptance matrix. The relevant inputs. The derivation of T for a single piezoelectric axial in DOFs of the device are divided into two groups: a group of actuator is further explained in Section 3, e.g., in Eq. 15.The DOFs that will be coupled with the spindle (subscript c) and measurements of the device then provide the following FRF a group of DOFs which will remain independent (subscript matrix: i). The uncoupled dynamic response of substructure D can then be described by the following: H = H T = H T m A,ii in D,ii in −1 − H ( H + H ) H T (5) D,ic D,cc S,cc D,ci in X F H H D,c D,c D,cc D,ci = H , where H = D D X F H H D,i D,i D,ic D,ii The goal now is to reconstruct H based on the known S,cc (2) dynamics of the assembly H and the device H . Equation 5 m D can be evaluated for any device, that fits the description of Here, X and X are vectors of frequency domain Eq. 2, as long as ( H + H ) is invertible, regardless D,c D,i D,cc S,cc displacements at coupled and independent coordinates, of the number of excitation inputs and sensors or their loca- respectively. Analogously, F and F are vectors of fre- tions. However, solving Eq. 5 in closed-form for H is D,c D,i S,cc quency domain forces at coupled and independent DOFs, only possible under certain conditions which are discussed in respectively. The detailed derivation of the device model is more detail in Section 2.2.2. A necessary condition is that the presented in Section 3. device offers equally many independent excitation inputs and 123 The International Journal of Advanced Manufacturing Technology sensors as components in F and X , respectively. Thus, Based on this spindle receptance model and the calibrated S,c S,c if the spindle receptance is considered to be a two-by-two dynamics of the device (see Section 3.1), Eq. 5 is evalu- receptance matrix, the device needs to offer two independent ated, and the predicted response H is compared to actual excitation inputs and two independent response sensors. measurements H at N frequency points ω through the fol- m i Based on this, two device prototypes are developed that lowing minimization loss function: are described in the following two subsections: The first one (Section 2.2.1) is simple and compact but does not offer suf- L ( H , H (θ )) = log (| H (ω )|)−log (| H (ω ,θ)|) m p m i p i 10 10 ficiently many excitation inputs for a closed-form spindle i =0 receptance reconstruction. Instead, a spindle reconstruction method based on a modal model of the spindle and gradient- +γ H (ω ) − H (ω ,θ) 1 m i p i based optimization is proposed. In the remainder of the i =0 document, this device is referred to as “one-stage device.” m +γ (max(0, −(ω − ω )) The other prototype (Section 2.2.2) is less compact but allows 2 k min k=0 to use a closed-form spindle reconstruction approach. The + max(0,ω − ω )) (7) k max term “two-stage device” shall refer to this device in the fol- lowing. with | H | and H representing the element-wise magnitude and phase of complex-valued matrix elements, respectively. 2.2.1 One-stage device and optimization-based spindle • also stands for Euclidean norm of matrices. In other receptance reconstruction words, the first two terms are the mean squared errors of the log magnitude and phase of the predicted and measured FRFs This prototype device is made from stainless steel and has the spindle interface of HSK A63 and the geometry shown in Fig. 2. It features a relatively compliant section in the middle, around which four axial piezoelectric actuators are arranged. Two opposing piezoelectric actuators are driven with voltages of equal magnitude and opposite polarity, hereby exciting bending modes of the structure. The device thus offers one independent excitation input for the X and one for the Y direction. To capture the dynamic response, six accelerometers—three in X and three in Y direction—are attached to the device. A modal model according to [23] (Eq. 3.74 in this refer- ence) is chosen to represent the spindle receptance matrix, as provided by the following: Rigid Coupling u u H (s) = (6) s + 2ζ ω s + ω k n,k n,k k=1 It assumes the linear behavior of the structure and inherently enforces the receptance matrix to be symmetric. Potential nonlinearities in the spindle receptance could be captured by making these modal parameters dependent on, e.g., static load or spindle speed, as is done by [24]. The number of Fig. 2 Structural arrangement of the intended machine tool with the modes m is a hyperparameter that needs to be determined in one-stage device. Highlighted piezoelectric actuators in gray are excited advance while the eigenfrequencies ω , damping ratios ζ , n,k k with opposite polarization to shake bending dynamics. Three indicated and mass-normalized mode shapes u are the optimization k accelerometers are meant to measure the lateral response (X direction) of the machine tool-device assembly parameters. Substructure D1 Substructure S (One-stage device) (Machine tool + Spindle) The International Journal of Advanced Manufacturing Technology Table 1 Overview of model Model parameter Reparametrization Initialization parameters, reparametrization, and initialization p k 1,k ω ω = e p = log(ω + (ω − ω )) n k 1,k min max min 2,k ζ ζ = e p = 0.01 k k 2,k ( p + jp ) 3,k,i 4,k,i u u = e p = p = 0.0 k,i k,i 3,k,i 4,k,i ω and ω are the bounds of the considered frequency range. m is a hyperparameter that sets the number min max of modes averaged over frequencies ω . The third term acts as a con- straint and encourages the eigenfrequencies to stay within the meaningful frequency range of (ω ,ω ). θ is the min max collection of optimization parameters, and γ and γ are 1 2 hyperparameters to balance the loss terms. For easier optimization, the model parameters are repara- metrized using the natural logarithm (for eigenfrequencies and damping ratios). This not only enforces them to remain positive but also allows easier exploration, i.e., to find the suitable order of magnitude more quickly. Similarly, the ele- ments of the mass-normalized mode shape u are expressed Rigid Coupling in terms of log magnitude and phase. An overview of optimization parameters and their reparametrizations and ini- tializations is given in Table 1. 1 1 PyTorch [25] is used for the implementation of the spindle model, device model, and loss function which allows using its automatic differentiation functionality to obtain gradients of the loss function. The Adam optimizer [26]isemployed to minimize the loss function using batch gradient descent. 2.2.2 Two-stage device and closed-form spindle receptance reconstruction X-direction Y-direction This device is made from stainless steel and has a spin- dle interface of HSK A63 similar to the one-stage device. Fig. 3 Structural arrangement of the intended machine tool with the However, at the cost of being less compact, the two-stage two-stage device. Highlighted piezoelectric actuators in gray and light device features a second stage of piezoelectric actuators red provide two independent excitations to machine tool-device assem- (Fig. 3). Again, two opposing piezoelectric actuators are dri- bly while the indicated accelerometers record the resulting lateral response. The indicated jamming bolts in the Y direction are used to ven with voltages of equal magnitude and opposite polarity. disturb symmetry in device dynamics in X and Y directions The two stages of piezoelectric actuators offer two inde- pendent excitation inputs per X and Y directions. Con- sequently, a closed-form spindle dynamics reconstruction method can be used, which is described next. Solving Eq. 5 to be invertible. The inverses of the latter two terms are for H yields the following: required in solving for H . In words, the invertibility S,cc S,cc of ( H T ) implies that the actuators embedded in the D,ci in −1 H = H T ( H T − H ) H − H (8) device need to be able to cause displacements in all DOFs S,cc D,ci in D,ii in m D,ic D,cc of X = X . Thus, if X =[X,θ ] , then two D,c S,c S,c x As H is expected to be symmetric, the off-diagonal com- actuators are needed that cause two linearly independent S,cc ponents are averaged, i.e., H = 0.5( H + H ). displacement vectors at the coupling point. Similarly, the S,cc,sym S,cc S,cc Equation 8 requires ( H T − H ), ( H T ), and H invertibility of H requires that any force vector F D,ii in m D,ci in D,ic D,ic D,c Substructure D2 Substructure S (Two-stage device) (Machine tool + Spindle) The International Journal of Advanced Manufacturing Technology Fig. 4 Translational and rotational components of the spindle dynamics second case, the Y -axis of the device was aligned with the X-axis of the of a GFMS Mikron HPM800U 5-axis milling machine in X direction machine. The two predictions are corrupted around different frequen- as estimated by the two-stage device in two different orientations and cies (approximately 1500 Hz and 1700 Hz), because the dynamics of a reference obtained as suggested in [27]. In one case, the X-axis of the two-stage device is different in X and Y directions the device was aligned with the X-axis of the machine while in the applied at the coupling point causes a unique displacement the X-axis of the machine, and once with the Y -axis of the vector X that can be captured by the embedded sensors. device aligned with the X-axis of the machine (i.e., a 90 D,i Hence, if F =− F , two sensors must be placed in rotation of the device between measurements). The same pro- S,c D,c different locations. Finally, the invertibility requirement of cedure should be carried out for measuring the Y -axis of the ( H T − H ) means that the measurements obtained machine. This essentially yields two estimates of the spin- D,ii in m from the device when mounted to the spindle should be dif- dle receptance from two different device dynamics (denoted ferent from measurements obtained from substructure D (the H and H ) per direction. Consequently, the S,cc1,sym S,cc2,sym device without HSK section) alone. Note that the above con- spindle receptance estimates obtained using Eq. 8 are cor- ditions must hold on a frequency-by-frequency basis in the rupted around different frequencies, as can be seen in Fig. 4. frequency range of interest. The FRF components of the two estimates are then com- The spindle receptance estimates resulting from Eq. 8 are bined by computing their weighted arithmetic mean. The corrupted by inevitable model mismatch, which becomes weights for measurements 1 and 2 and are defined as particularly pronounced around eigenfrequencies of the device where the term that needs to be inverted may become |σ (ω)| 1,ij r (ω) = 1 − ill-conditioned. Reconstruction errors can be reduced if the 1,ij |σ (ω)|+|σ (ω)| 1,ij 2,ij eigenfrequencies of the device are chosen to be outside the |σ (ω)| 2,ij frequency range of interest or if the device model can be r (ω) = 1 − (9) 2,ij |σ (ω)|+|σ (ω)| 1,ij 2,ij determined with extremely high accuracy. If neither of these requirements can be satisfied, a remaining solution is to repeat the spindle receptance measurement with two different where σ (ω) is the sliding window standard deviation (con- k,ij device dynamics, i.e., different in terms of their eigenfre- sidering a window length of 2 P + 1 samples) of the FRF quencies. This approach is chosen to obtain accurate spindle component ij of measurement k at frequency point ω com- receptance estimates with the available prototype. The device puted as follows: dynamics in the Y direction are altered by jamming a pair of bolts in the lower piezo stage, as shown in Fig. 3. The measurement of the spindle receptance should be then ¯ 2 σ (ω ) = (H (ω ) − H ) (10) k,ij l k,ij l+ p k,ij performed once with the X-axis of the device aligned with 2 P + 1 p=− P 123 The International Journal of Advanced Manufacturing Technology , coupled machine tool-device dynamic behavior according to Eq. 4. Due to the design of the prototype devices described in Sections 2.2.1 and 2.2.2, its lateral dynamics in X and Y directions are expected to be independent of each other (i.e., no axis cross-coupling). The models of the device 2 2 dynamics presented in the following are thus restricted to uni-directional translational and rotational deformations. The , , HSK-A63 interface is not part of the device models, as it is , , 1 1 1 1 considered to be part of the machine tool. In Section 3.1, it is suggested that utilizing a simplified , , 1 1 1 1 Timoshenko beam model could yield an adequately pre- , , cise model for the one-stage device and optimization-based spindle receptance reconstruction approach. The two-stage 3 3 device, on the other hand, involves more components and complexities, and hence, developing a physics-based device model through modeling and calibration could be an exten- sive task. To provide an alternative solution, Section 3.2 presents an experimental approach based on inverse RCSA 4 4 , for obtaining device models. 5 5 3.1 Physics-based modeling and calibration The receptance of the one-stage prototype device is modeled through sequential receptance coupling of cylinders under axial or lateral load, as detailed in Section 3.1.2.The Timo- shenko beam model is used to compute the receptance of a single cylinder subject to lateral forces and bending moments at its ends. For the axial receptance of a uniform rod, the ana- lytical solutions provided in [28] are used. The cylindrical piezoelectric axial actuators embedded in the device are mod- Fig. 5 Dynamics model of the one-stage device: Cylinders (drawn as eled as mere elastic structures. The converse piezoelectric rectangles) subject to lateral or axial load are assembled through rigid effect is modeled by a pair of external forces as described in and elastic receptance coupling. External force inputs (blue) model the Section 3.1.1, following the approach taken in [29]. piezoelectric effect of the applied voltage. Displacement outputs are denoted in red 3.1.1 Piezoelectric actuator Before computing the weighted average, the two mea- surements are smoothed using moving median filtering on Let the axial receptance of a linear elastic rod be given magnitude and phase. Finally, the resulting spindle recep- through the following receptance matrix: tance is computed as H = r H + S,cc,av,ij 1,ij S,cc1,sym,ij r H . The FRFs of the device are obtained as 2,ij S,cc2,sym,ij Z H H F 1 11 12 z1 = (11) described in Section 3.2. Z H H F 2 21 22 z2 The elongation of a piezoelectric material due to the volt- 3 Bending dynamics of the measurement age V applied across its end faces can be described by devices This section is devoted to obtaining the required dynamics z1 Z − Z = H − H H − H + d V (12) 2 1 21 11 22 12 33 of the two device configurations as they are involved in the z2 123 The International Journal of Advanced Manufacturing Technology where the force-induced portion of the deformation is based 3.1.3 Model calibration on the axial receptance according to Eq. 11 and d is the piezoelectric coefficient. The voltage-induced portion of the To capture the dynamics of the device alone (i.e., not attached elongation is now replaced by a pair of external forces of to any machine tool), it is hung from an elastic string to equal magnitude F and opposite directions: simulate free-free boundary conditions. FRF measurements ext are then obtained from the actuators and sensors embedded in the device as well as from tap testing on two points on the F + F z1 ext Z − Z = H − H H − H (13) 2 1 21 11 22 12 tool-holder part. F − F z2 ext A Timoshenko beam-based model of the tool-holder taper is added to the device model through receptance coupling, as Comparing the above two equations yields an expression it is considered to be part of the machine tool and is thus not for the magnitude of the external force in terms of applied included in the device model. The device model is then cal- voltage and receptance of the piezoelectric material: ibrated by tuning model parameters using genetic algorithm optimization, such that the obtained measurements match d V F = (14) ext the FRFs predicted by the device model. All of the afore- H + H − H − H 21 12 11 22 , c ) as well as the outer mentioned contact parameters (k i i diameter and Young’s modulus of the center bolt are consid- The mapping from actuator inputs (a single excitation volt- ered optimization parameters. The properties of the center age in this case) to external forces reads as follows: bolt are tuned to take the effect of the two/four piezo stacks into account which is not considered in this planar model. F 1 33 z1,ext = T V = V in F −1 H + H − H − H z2,ext 21 12 11 22 3.2 Experimental modeling through inverse RCSA (15) This approach aims to obtain all required components of 3.1.2 Assembled model through coupling substructures the device FRFs through measurements. The obstacle here is applying rotational moments at the coupling point with An overview of the dynamics models of the one-stage device the spindle, which is not practical in reality. To circum- is given in Fig. 5. The cylinders are drawn as rectangles; vent this issue, the top part of the device is modeled using their center axes as dash-dotted lines. Solid lines denote rigid a Timoshenko beam model and is considered known. The couplings whereas linear and rotational elastic couplings are FRF measurements are taken from the device in free-free indicated by the respective pictograms. The latter has been condition using the embedded actuators and sensors on the introduced between the piezoelectric elements and the neigh- unknown part of the device, as well as through impacting boring components (k , c ). They are supposed to model the 1 1 and measuring on the known part of the device (Fig. 7). The effect of the 0.1 mm thin copper electrodes and a 0.07-mm FRFs of the unknown part of the device can then be obtained thin plastic insulation sheet which are found in these locations using inverse RCSA. Finally, part of the subtracted structure in the physical prototype. Axial forces and displacements at is added again to get the model of the device up to the desired the ends of the two piezo stacks are mapped to moments and coupling location. A graphical overview of the method is rotational displacements through linear mapping. given in Fig. 6. Elastic couplings are further introduced between this For obtaining the measurements, the device was again assembly of piezo stacks and the surrounding frame (k , c 2 2 hung from an elastic string to simulate free-free boundary and k , c ). They should take into account any local defor- conditions, as shown in Fig. 7. 3 3 mations and contact stiffnesses. Finally, linear and rotational spring-damper systems are introduced where the physical device is bolted together (k , c , k , c ). The free inputs and 4 4 5 5 4 Experimental validations outputs of the device model are indicated in blue and red, respectively. For the one-stage device, they correspond to The two versions of the measurement device are used to esti- F and X from Eq. 2 as follows: D,i D,i mate the spindle receptance of a GFMS Mikron HPM800U 5-axis milling machine in the setup shown in Fig. 8.An exponential chirp excitation ranging from approximately 80 F = F F F F (16) D,i D,i0 D,i1 D,i2 D,i3 Hz to 6 kHz with a voltage amplitude slightly below 100 X = X X X (17) D,i D,i0 D,i1 D,i2 V is used for both devices. The excitation and response 123 The International Journal of Advanced Manufacturing Technology F X Si,0 Si,0 known Di,0 known X = X Di,0 Si,1 Si,1 Di,0 Inverse RCSA known Di,1 Di,1 Di,2 Fig. 6 Device dynamics identification method based on inverse RCSA, where the FRFs of unknown substructures are determined by subtracting theoretical models of known parts from the measurements of the intact device signals are sampled at 51.2 kHz using a National Instru- the resulting tooltip dynamics using RCSA to impact test- ments data acquisition card type 9234. PCB Piezotronics ing measurements. Seven different tool-holder combinations accelerometers of type 352C22 are employed to capture the response. The measurements are repeated 100 times to aver- age out measurement noise. The number of modes of the modal spindle model used with the one-stage device and the optimization-based spindle receptance reconstruction is chosen to be m = 10 as a hyperparameter. The resulting modal parameters can be found in Appendix A.For thetwo- stage device, measurements are conducted as multiple SIMO (single input multi output) experiments, using one excitation input at a time and setting the other voltage input to zero by short-circuiting the electrodes of the piezoelectric actuators. 4.1 Validations of estimated spindle dynamics versus manual impact testing The estimated spindle dynamics by the two device layouts are compared against reference values in Fig. 9.The refer- ence receptance is obtained through an inverse receptance coupling approach as suggested by Namazi et al. [27]. In this method, a cylindrical dummy holder is mounted to the machine, and FRFs are measured at certain locations through impact testing. This indirect measurement method is proposed since the excitation of rotational dynamics by a hammer is not straightforward. In the following, the mea- surements of the machine’s dynamics at the spindle flange are compared against each other. 4.2 Validations through tooltip dynamics As mentioned earlier, the structural dynamics at the tooltip determine the boundaries of stable process conditions. Hence, it is valuable to evaluate the validity of the reconstructed spindle dynamics from the proposed devices by comparing Fig. 7 The two-stage device during structural identification. To simu- late free-free boundary conditions, it is hung from an elastic string 123 The International Journal of Advanced Manufacturing Technology Fig. 8 The prototype devices mounted to a GFMS Mikron HPM800U 5-axis machining center. Left: one-stage device. Right: two-stage device are selected for this validation purpose. The details of these ibility and equilibrium conditions respectively, the coupled combinations are provided in Table 2. The dynamics of the dynamics of the system at the tooltip can be computed as tooling systems are obtained through a finite element mod- follows: eling approach based on Timoshenko beam elements with −1 considering a non-uniformly distributed contact flexibility H = H − H ( H + H ) H (18) ST ,11 T ,11 T ,12 S,33 T ,22 T ,21 at the tool-holder interface. More details and corresponding values of the contact parameters can be found in [13]. Further details of substructure coupling through the RCSA Considering the substructuring scenario presented in method can be found in publications of Schmitz et al. [30, Fig. 10 with X = X and F + F = 0 as compat- T ,2 S,3 T ,2 S,3 31]. Fig. 9 Translational and rotational components of the spindle dynamics of a GFMS Mikron HPM800U 5-axis milling machine in X and Y directions as estimated by the one-stage device, the two-stage device, and the reference obtained through iRCSA 123 The International Journal of Advanced Manufacturing Technology Table 2 Tooling systems (S.L. # Holder Tool S.L. (mm) Picture indicates the stickout length of the cutting tool) 1 Zürn HSK-A63 63.11.20.2 Voha 12032456120 70.50 Interface: Collet ER32 4 flutes 2 REGO-FIX PG25x100H Diameter: 12.0 mm 65.90 Interface: powRgrip PG25 Length: 110.5 mm 3 REGO-FIX PG25x100H 60.90 Interface: powRgrip PG25 4 Schunk 208123 50161717 62.60 Interface: Thermal shrink fit 5 Zürn HSK-A63 11.16.23 Voha 2002805 Interface: Collet ER25 4 flutes 60.07 Diameter: 16.0 mm Length: 93.3 mm 6 Long insert tool 5 cutters − Diameter: 40.0 mm 7 Short insert tool 5 cutters − Diameter: 50.0 mm Figure 11 shows tooltip receptance measurements for the 4.3 Validations of predicted stability boundaries tool-holder combinations described in Table 2 along with the against experimental observations predictions using on the spindle receptance measurements from the one-stage device, the two-stage device, and the ref- Experimental observations of stability states and correspond- erence dynamics through iRCSA. ing chatter frequencies are used to verify the model-based prediction of SLDs through the zero-order approximation method by Altintas and Budak [3]. The required tooltip dynamics are taken from the RCSA coupling presented in the preceding section. The validation cuts are collected through experimental cutting tests for the first two tool-holder combi- nations from Table 2 on a block of Al6082 aluminum. Table 3 summarizes the corresponding process parameters. In the prediction of stability charts, the experimentally calibrated values of 902 MPa and 243 MPa are assumed for the tangen- tial and radial cutting force coefficient, respectively. An acceptable overall agreement between the three pre- dictions and the validation data can be observed for the first validation case in Fig. 12a. In the second case (Fig. 12b), the predictions based on the reference spindle receptance Rigid Coupling and the estimate from the two-stage device are in agree- ment, but the predicted critical depth of cut based on the spindle receptance estimate from the one-stage device is slightly higher. This is due to an inaccuracy in the magni- tude of the corresponding tooltip receptance in Y direction (Fig. 11b), which is underestimated for frequencies around the dominant mode (2100–2300 Hz). It is worth noting that the SLDs generated using the measured tooltip FRFs show slight deviations from those produced by the predicted FRFs. Fig. 10 Coupling scenario of the machine tool structure and tooling However, this deviation is not necessarily positive in terms system. The model for tooling systems is obtained from the method of improved prediction accuracy of the stability borders ver- presented in [13] Substructure T Substructure S (Tool-holder assembly) (Machine tool + Spindle) The International Journal of Advanced Manufacturing Technology Fig. 11 Tooltip receptance predictions and measurement for tool- using the method presented in [27]. These spindle receptances are holder combinations #1 to #7 in Table 2 correspond to a to g, rigidly coupled with Timoshenko-beam-based tooling models to obtain respectively. Predictions are based on spindle receptance estimations tooltip predictions from the one-stage device, two-stage device, and a reference obtained 123 The International Journal of Advanced Manufacturing Technology Table 3 Process parameters Tooling Feed Milling Radial depth Feed rate used for the validation cuts with Case system direction Strategy of cut (mm) (mm/tooth) two tool-holder combinations (according to Table 2)ona 11 Y + Slotting 12.0 0.05 block of Al6082 aluminum 22 X+ Up-milling 8.4 0.05 sus the validation cuts. This raises the possibility that the more, the spindle receptance reconstruction method could direct measurement of tooltip FRFs using an impact hammer be expanded to capture axis cross-couplings, which involves and accelerometer may be compromised. The imprecision the response of the spindle in the X direction to the excitation may stem from misalignment in the direction and location of in the Y direction and vice versa. hammer impacts or the accelerometer, particularly given the fluted geometry of the tooltip. 5 Conclusion This study introduced two prototypes of measurement devices and their identification methods to automatically identify machine tool dynamics at the spindle interface. The two devices differ in the number of independent piezoelec- tric actuators and embedded acceleration sensors, as well as in their compactness. Both approaches produced satisfactory results. The estimation of spindle dynamics using the one-stage device provides a compact measurement system with accept- able measurement accuracy. In the underlying optimization- based reconstruction method, like any optimization problem, attention must be given to the convergence to local min- (a) imums, such as when translational and rotational FRFs compensate for each other. Further inclusion of physical con- straints or prior knowledge can be helpful for the convergence of the optimization to global solutions. The two-stage device and the closed-form reconstruc- tion method for spindle receptance estimation led to faster, unique, and repeatable results. However, the device com- pactness is compromised due to the additional stage of piezoelectric actuators. Future work could explore a more sophisticated arrangement of piezoelectric actuators, such as the Stewart-platform arrangement [32], which offers independent excitation inputs while maintaining a compact design. Improving the accuracy of the device model can further enhance the precision of the reconstructed spindle dynamics, which encourages arrangements that can be more reliably modeled in the design stage. The two proposed measurement devices enabled the iden- tification of machine tool receptance with high repeatability and minimal human effort, as only mounting the device on (b) the machine tool is required. Additionally, the approach is Fig. 12 Predicted stability boundaries and experimental chatter obser- suitable for measuring the machine tool dynamics in various vations for tool-holder combination #1 in a and #2 in b from Table 2 locations in the working space since the device can be easily using process parameters according to Table 3 on a block of Al6082 moved around without requiring external support. Further- aluminum 123 The International Journal of Advanced Manufacturing Technology Appendix A: Modal parameters of estimated spindle receptances Table 4 Modal parameters of k ω [rad/s] ζ [−] u [ s/kg] u [rad s/kgm ] n,k k k,0 k,1 the spindle receptance in X direction as estimated by the 1 1924.67 0.003 0.01134-0.00145j −0.0271+0.0310j one-stage device and 2 4518.64 0.163 0.17878+0.025827j 1.9041+0.89598j optimization-based identification method 3 8562.86 0.028 0.23317−0.079048j 0.80471−0.085383j 4 9125.85 0.461 0.14483−0.72793j 2.7327−9.7517j 5 10,890.80 0.165 0.67985−0.64180j 2.2236−6.5233j 6 12,421.76 0.131 1.0155−0.10138j 5.3805−4.5614j 7 14,057.43 0.155 1.0305+0.35473j 9.6915−3.0552j 8 21,090.84 0.231 0.96087+0.61073j 0.31771+0.63735j 9 22,391.05 0.548 1.3746+1.3535j 36.942+10.874j 10 22,808.70 0.224 −0.849+0.97204j −3.3365+18.436j Table 5 Modal parameters of √ k ω [rad/s] ζ [−] u [ s/kg] u [rad s/kgm ] n,k k k,0 k,1 the spindle receptance in Y direction as estimated by the 1 2239.41 0.113 0.03061−0.044494j 0.0235+0.00492j one-stage device and 2 4584.90 0.111 0.1209−0.03875j 1.8830−0.32124j optimization-based identification method 3 8107.34 0.141 0.29544+0.0776j −2.9079−3.5816j 4 9437.61 0.031 0.1990−0.09720j 0.8738−1.8657j 5 10,271.24 0.109 0.28643+0.1873j 0.9037−2.8547j 6 15,633.82 0.251 1.2377−0.46629j 8.3055−0.17174j 7 17,837.63 0.079 0.43768+0.15219j 4.8724−5.0213j 8 20,598.82 0.162 0.7702+0.12895j 0.11743−13.038j 9 21,461.37 0.441 0.75704+1.3509j 36.08+15.436j 10 23,434.09 0.023 0.2007+0.042563j 0.20581+0.29781j 123 The International Journal of Advanced Manufacturing Technology Author Contributions VOAA: conceptualization, methodology, super- 14. Akbari VOA, Mohammadi Y, Kuffa M, Wegener K (2023) Identifi- vision, final draft editions. CS: methodology, analysis, writing original cation of in-process machine tool dynamics using forced vibrations draft, experimental setup, data curation. MK and KW: supervision, in milling process. Int J Mech Sci 239:107887 review, and funding acquisition. 15. Liu X, Cheng K (2005) Modelling the machining dynamics of peripheral milling. Int J Mach Tools Manuf 45(11):1301–1320 Funding Open access funding provided by Swiss Federal Institute of 16. Montevecchi F, Grossi N, Scippa A, Campatelli G (2017) Two- points-based receptance coupling method for tool-tip dynamics Technology Zurich. This research was financially supported by Inno- prediction. Mach Sci Technol 21(1):136–156 suisse, the Swiss Innovation Agency (grant 32334.1 IP-ICT). 17. Namazi M, Altintas Y, Abe T, Rajapakse N (2007) Modeling and identification of tool holder-spindle interface dynamics. Int J Mach Declarations Tools Manuf 47(9):1333–1341 18. Akbari VOA, Kuffa M, Wegener K (2023) Physics-informed Bayesian machine learning for probabilistic inference and refine- Conflict of interest The authors declare no competing interests. ment of milling stability predictions. CIRP J Manuf Sci Technol 45:225–239 Open Access This article is licensed under a Creative Commons 19. Wegener K, Weikert S, Mayr J, Maier M, Ali Akbari VO, Postel M Attribution 4.0 International License, which permits use, sharing, adap- (2021) Operator integrated-concept for manufacturing intelligence. tation, distribution and reproduction in any medium or format, as J Mach Eng 21 long as you give appropriate credit to the original author(s) and the 20. Postel M, Bugdayci NB, Monnin J, Kuster F, Wegener K (2018) source, provide a link to the Creative Commons licence, and indi- Improved stability predictions in milling through more realistic cate if changes were made. The images or other third party material load conditions. Proc CIRP 77:102–105 in this article are included in the article’s Creative Commons licence, 21. Postel M, Candia N, Bugdayci B, Kuster F, Wegener K (2019) unless indicated otherwise in a credit line to the material. If material Development and application of an automated impulse hammer is not included in the article’s Creative Commons licence and your for improved analysis of five-axis CNC machine dynamics and intended use is not permitted by statutory regulation or exceeds the enhanced stability chart prediction. Int J Mechatron Manuf Syst permitted use, you will need to obtain permission directly from the copy- 12(3–4):318–343 right holder. To view a copy of this licence, visit http://creativecomm 22. Ferreira J, Ewins D (1996) Nonlinear receptance coupling approach ons.org/licenses/by/4.0/. based on describing functions. In: Proceedings-SPIE the interna- tional society for optical engineering, pp. 1034–1040 23. 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Journal

The International Journal of Advanced Manufacturing Technology – Springer Journals

Published: Feb 1, 2024

Keywords: Chatter stability; Machine tool dynamics; Piezoelectric actuators; Structural coupling; Inverse RCSA

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