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Prediction of spindle dynamics in milling by sub-structure coupling
International Journal of Machine Tools & Manufacture 46 (2006) 243–251 www.elsevier.com/locate/ijmactool
Prediction of spindle dynamics in milling by sub-structure coupling Mohammad R. Movahhedy*, Javad M. Gerami Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Azadi Ave., Tehran, Iran Received 24 February 2005; accepted 26 May 2005 Available online 8 August 2005
Abstract The Stability of machining process depends on the dynamics of the machine tool, among other things. However, the dynamics of the machine tool changes when the tool is changed. To avoid the need for repeating the measurements, sub-structuring analysis may be used to couple the tool and spindle frequency response functions. A major difficulty in this approach is the determination of joint stiffness and damping between the two sub-structures. In particular, the measurement of rotational responses (RDOFs) at joints is a difficult task. In this research, a simple joint model that accounts for RDOFs is proposed. It is shown that this model avoids RDOF measurement while taking into account the bending modes. An optimization method based on genetic algorithm is employed to find parameters of the joint model. Receptance coupling analysis is used to couple the machine tool and tool FRFs obtained from experiment and FE, respectively. The RC based response obtained in this way is compared with experimental FRF which shows good agreement and confirms that the joint model has been successful in predicting the tool bending modes. q 2005 Elsevier Ltd. All rights reserved. Keywords: Sub-structure coupling; Machining stability; Chatter vibration; Genetic algorithm
1. Introduction The rising trend in the use of high speed machining in aerospace, automotive and other industries has presented new challenges for the machine tool manufacturers and users. High speed machining results in higher material removal rates and better surface quality, but higher machine tool strength and higher precision are required. In particular, the dynamics of the machine tool has a determining effect on the precision and removal rate of the machining process due to the possibility of occurrence of self-excited chatter vibration. For example, in machining of turbo jet engines, long and flexible tools are used which are prone to the chatter occurrence. If cutting parameters such as depth of cut and cutting speed are not selected appropriately, machining process can become unstable resulting in serious damage to machine tool spindle and scrapping of expensive workpieces. It is well established that the occurrence of chatter is dependent on three factors: cutting conditions, work piece * Corresponding author. Tel.: C98 21 616 5505; fax: C98 21 600 0021. E-mail address: [email protected] (M.R. Movahhedy).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.05.026
material properties and the dynamics of machine tool system. The stability lobe diagram, which predicts stable sets of machining parameters, is commonly used to select best parameters for avoiding chatter. Knowledge of the machine tool dynamics, i.e. its frequency response (FRF), at the tool tip is an essential requirement for obtaining stability lobe diagrams [1,2]. Such system responses are traditionally obtained by performing modal tests on the machine tool or by numerical modeling. Experimental FRF is the most reliable method for systems with complex geometries, but is expensive and time consuming, particularly if it has to be repeated for every setup. Analytical and numerical FRF, on the other hand, will inevitably contain approximations and should be carefully validated if system is complex and comprised of many parts assembled together. However, it may be used more reliably for simpler, e.g. single piece, systems. In a milling machine, the response of the machine at the tool tip is mostly affected by the dynamics of spindle, supported on its bearings, and the cutting tool, which is usually the most flexible part of the system. The cutting tool is usually attached to the spindle through a standard tool holder. Due to complexity of spindle geometry and the effects of bearing stiffness and damping, it is customary that the response of the combined spindle-tool holder-tool
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system is obtained from modal testing. However, when a new tool is attached to the spindle, the dynamics of the system changes, and this change may be significant from one setup to another, because the tool is the most flexible part of the assembly and is more prone to vibration. As a result, the modal test may have to be repeated for every new tool setup. However, since it is only the tool that changes in the system, it is possible to obtain the dynamic response of the spindle system—without the tool—only once, and then combine it with the response of the tool by coupling of the two substructures. The response of various tools may be generated either experimentally or by numerical modeling. Due to relative simplicity of the cutting tool structure, it is expected that numerical modeling can yield reliable frequency response functions (FRF) for various cutting tools. Tool holder may be considered as part of spindle, tool, or as a separate substructure. In structural dynamics, the coupling of responses of two separate systems to determine the response of their assembly is known as receptance coupling method (RC). The RC method can significantly reduce the cost of obtaining system response for various setups. However, a major problem in using this method is in identification of stiffness and damping properties introduced to the assembly at joints between substructures. For example, Schmitt et al. [3] combined the analytical tool response and coupled it with spindle FRF, but they adjusted joint parameters in a way that the results of analysis match those of the experiment. Another obstacle in using the RC method is in measuring of responses due to rotational degrees of freedom (RDOF). In the structures which carry bending loads, RDOFs have a significant effect on the dynamics of the structure [4]. However, measurement of RDOF responses is technically difficult. Common methods use a finite difference scheme to obtain the RDOF responses from responses of two linear accelerometers. New laser instruments for angle measurement are recently introduced, but are usually expensive and may not cover all the required data. However, the inclusion of RDOF responses is essential for successful coupling of substructure FRFs [5]. While the rotational FRFs for the tool may be readily generated numerically, the corresponding experimental FRFs are not available for the spindle system. This research proposes a joint model which avoids RDOF measurements for prediction of machine tool assembly FRF. In this model, two linear joint elements are used and the applicability of the model in successful coupling is numerically verified. Using this model, the spindle-tool assembly FRF may be obtained without using rotational FRFs. The model is then applied to real machine tool cases and an optimization method based on genetic algorithm is used to identify joint parameters between tool and spindle. The results are verified through comparison with experimental results.
2. Receptance coupling theory The dynamics of a complex structure can be obtained by combining the dynamics of its substructures either in the modal space or in the response space. The former approach is called component mode synthesis and usually involves approximations due to curve fitting errors. Besides, the effects of high frequency modes that are truncated from analysis are not included. The latter approach is called receptance-coupling (RC) method in which, the effects of high frequency modes are inherently present in the data, but the presence of noise in the experimental data may hinder the success of the coupling procedure. In the RC method, the FRFs of the components of a structure are independently obtained, and are then coupled together analytically by applying the compatibility and equilibrium conditions. Since the components are assembled through mechanical joints that may have mass and stiffness, the dynamics of the joint should be included in the assembled system. If the joint properties are known, this amounts to a direct problem of combining the data [6]. However, usually the joint properties are not known and cannot be measured independently. In such cases, an inverse problem should be solved in which the joint properties are identified from the responses of components and assembly. This is called joint identification problem. Let two substructures with given frequency responses h and h 0 , are coupled together to form a structure whose response is denoted by g. Also, let the degrees of freedom of the substructures are divided into those which are at the joint and those away from the joint, and denote these by subscripts j and n, respectively, and reorder the DOFs such that similar DOFs are grouped together. For the two components, the dynamics equation can be written as #( ) " fn hnn hnj xn Z ; xj hjn hjj fj C F J (1) #( ) " 0 0 fn 0 hn n hn 0 j 0 xn 0 Z 0 xj 0 hj 0 n 0 hj 0 j 0 fj 0 C F J In this equation, for example, hnj shows the response at a non-joint dof n due to force at a joint dof j. For the assembled structure, we have: 8 9 2 38 f 9 n > > 0 0 g g g g x > > > nn nj nj nn n > > > > > > > > > = < 4 gj 0 n gj 0 j gj 0 j 0 gj 0 n 0 5 > xj 0 > fj 0 > > > > > > > > > > > : ; gn 0 n gn 0 j gn 0 j 0 gn 0 n 0 : f 0 ; xn 0 n
Equilibrium condition requires that F j Z KF j
0
(3)
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and compatibility condition requires that Xj0 K Xj Z H j F j
(4)
where Hj is the receptance matrix of joint, defined as H J Z ½K C juCK1
(5)
in which K and C are stiffness and damping matrices of structural joints. Each element of the FRF matrix in Eq. (2) is an individual frequency response function for corresponding excitation and response positions. We can obtain these FRFs by substitution of Eqs. (1) and (3) in Eq. (4). For example, a non-joint DOF of structure is obtained as a function of substructures’ and joint FRFs as: gnn Z hnn K hnj ðH J C hjj C hj0 j 0 ÞK1 hjn
2
where Annff means linear response due to linear excitation f, Annfm means linear response due to rotational excitation m, and Annmf means rotation due to linear excitation f, etc. [6].
3. Tool-spindle joint model In this section, we will discuss the issue of coupling the FRF of a cutting tool with the FRF of spindle on a milling machine. It is assumed that the dynamics of the machine mainly contributed by its spindle and the other parts of its structure are relatively rigid. Furthermore, the tool holder is considered as part of the spindle assembly for convenience. Due to the complex geometry of the spindle, and since it does not change considerably over time, it is preferred to obtain the response of this substructure by modal testing. The tool, on the other hand, has a much simpler geometry and its response may be obtained analytically or numerically, e.g. from an FE analysis. The difficulty here is that for the spindle, only the first element of the matrices given in Eq. (7) is readily available from modal test. For tool, these elements are readily available from FE analysis. It is shown [6] that if rotational DOFs are not included in the RC equations, the coupling exercise is likely to fail. Another important issue in RC is proper modeling of the tool-spindle joint. The joint used for connection between the two carries both a force and a moment. To model the flexibility of such a joint, a combination of a linear spring and a rotational spring may be used. Here again, the rotational DOFs are involved. Here, instead, a couple of parallel linear springs are used which together can carry both a force and a moment. To verify the equivalence of these two models, a numerical exercise using FE analysis is carried out. In this exercise, a spindle-tool system is
3
3 2
(6)
If rotational DOFs are taken into account, each of the FRFs in Eq. (6) is indeed a matrix composed of four response functions such as " # Annff Annfm ½Ann Z ; (7) Annmf Annmm
5
4
Model A
Model B
1
1
Fig. 1. Model A with rotational and linear joint elements versus Model B with two linear joint elements.
considered in which the joint between the two components is modeled in two different ways; model A connects the two components by a linear and a rotational spring, while model B connects the two through a couple of parallel linear springs (Fig. 1). Stiffness and damping parameters of the two sets of springs are calculated in a way that both joint models introduce similar modes to the assembled system. Table 1 shows the first four frequencies of the assembled systems for both models. An examination of the corresponding mode shapes shows that the second mode at 315 Hz is the first tool mode, the fourth mode at 1716 Hz is the first spindle mode, and the first and third modes at 194 Hz and 948 Hz are the contributions of the joint springs. The mode at 194 Hz represents the rotational motion at the joint, while the mode at 948 Hz represents linear displacement at the joints. Fig. 2 shows the two joint mode shapes for both models exhibiting equivalent shapes. Next, the RC method (Eq. 6) is used to calculate the response of the assembled system from the component FRFs for the two joint models. For model A, all the linear and rotational DOFs of the components are included in RC equations, while for model B, only linear DOFs are considered. Fig. 3 shows that the FRFs of the two models are similar and exhibit all the modes. However, if only linear DOFs are considered for model A, it will not exhibit the mode at 194 Hz that is related to the joint rotational stiffness. This is shown in Fig. 4, where such response is compared with the assembly response computed directly Table 1 The first four natural frequencies of tool-spindle assembly Natural frequencies in Hz Model A Model B
194.25 194.22
314.99 315.30
947.92 947.95
1716.1 1715.8
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Fig. 2. First and third tool-spindle assembly mode shapes at 194 and 947 Hz.
from FE analysis. On the other hand, model B exhibits the first mode because of the moment effect of the two linear springs. From this simple exercise it is concluded that when the linear assembly response at the tool tip is required, e.g. for stability analysis of the machining process, the exclusion of rotational DOFs may result in the disappearance of some modes with significant effects on the stability analysis. However, using model B for the joint compensates this shortcoming and removes the need for measurement of the rotational responses of spindle sub-structure. Finally, it should be mentioned that the assembly response is very sensitive to the joint parameters, and a change in these parameters will considerably shift the peaks affected by joint dynamics.
this practice will involve inverting of matrices with very small elements and low ranks, and is very sensitive to noise and errors that are inherent to the experimental data. For instance, if an error of only a few percent is introduced to the data, the identified joint properties may show much larger difference, e.g. 50%, from the actual levels. Researchers have presented procedures that avoid matrix inversion as much as possible to reduce numerical errors (see e.g. [7,8]).
4. Joint identification In this section, based on the joint model introduced and verified above, the RC method is used for coupling of the tool and spindle system. The first step for this purpose is to obtain the joint properties. If in Eq. (6) or similar equations derived from Eq. (2), the assembly and the components’ FRFs are known, the joint receptance matrix (Hj) can be obtained by algebraically inverting the equations. However,
Fig. 3. Predicted response of models A and B by RC analysis.
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Fig. 4. The effect of elimination of RDOF response in FRF prediction of Model A.
Alternatively, Eq. 6 can be solved as a nonlinear optimization problem for the joint properties. The optimization method finds the appropriate joint parameters that minimize the difference between RC-based assembly FRF and the corresponding experimental FRF. The objective function may be constructed as the following vector: ( ) Reðgp K gm Þ jgj Z (8) Imðgp K gm Þ in which gp and gm are predicted (from Eq. (6)) and measured receptances of the assembly. Euclidian norm of vector g is used as the objective function. Since the response values are complex, it is divided into real and imaginary parts. Also response values may be predicted and measured at various joint and non-joint DOFs and at a vast range of frequency points to expand the data space. In this work, the genetic algorithm (GA) search method is used to minimize the objective function. GA is a global search method which acts on the space of results [9]. All variables are coded with binary strings. A population of acceptable values is generated on the space of results. Each individual in the population is a binary string that can be decoded into real space. Individuals are evaluated with respect to their objective function values. Best individuals which minimize objective function are selected to generate a new population. Next population are generated by cross over, mutation and reinsertion of selected individuals. New individuals are evaluated and the best of them are selected to generate new population and the loop is continued to minimize objective function. The GA is preferred over other classic optimization algorithms because the latter
algorithms are generally very sensitive to the starting point, and quite often converge to local minimum and stop prematurely, while GA is a global search method that rarely converges to local minima. In addition, it is not necessary to specify a start point for GA; the space of acceptable results is enough for GA to find the minimum of a function. The parameters that are binary coded for the joint identification problem are the joint parameters, i.e. the stiffness and damping coefficients at the joints. The initial population consists of member sets of randomly generated coefficients within the specified range. For every member of this population, the objective function is calculated and evaluated. Next, 10% of the population that have yielded highest function values are erased and the rest of the population are crossed over at a probability of 0.7. In other words, 70% of the new population is generated by dividing the binary strings into parts and putting various parts together. Furthermore, 5% of the population is changed by mutation, in which a digit in the target string is switched from 0 to 1 or vice versa. Some of the member sets from the old population are also included. The new combinations of joint parameters formed by such crossover, mutation and reinsertion forms the new population. The function values are calculated for this new population and the process is repeated until it converges to the minimum solution. Evaluation of convergence rate is carried out at intervals and if the solutions are not satisfactory, the process is started over with a new set of population members. The run time of the algorithm depends on the number of population members, the string length and number of iterations. Increasing the number of members
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Fig. 5. Frequency response of spindle and tool-spindle assembly at the tool tip.
reduces the risk of convergence to a local minimum. In most problems, this number is set between 30 and 100. A longer string results in a more accurate solution. Also, increasing the number of members reduces the number of iterations in general.
In this work, the genetic algorithm module of MATLAB is used for finding joint parameters that minimize Eq. (8). The time to convergence depends on number of equations involved and the frequency range of interest. In the problem considered here, at least four RC equations were written for
Fig. 6. Frequency response of spindle and tool-spindle assembly on the tool-holder.
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Fig. 7. GA convergence to minimum solution.
Table 2 identified joint parameters k1 (N/m)
k2 (N/m)
c1 (N s/m)
c2 (N s/m)
6.31!108
9.82!1011
7.86
6.13
each frequency point in the specified range, and the norm of error at all frequencies was calculated as the objective function. Typically, the algorithm converged to the optimum solution within around 30 iterations, and in about 10–20 min on a Pentium IV processor.
5. Applications and verification To apply the above procedures, two experimental tests have been carried out. In the first experiment, a vertical 3-axis milling machine and a 19.05 mm diameter end
249
mill were used. In this test, first, a mechanical chuck is attached to the spindle, without the tool and the spindletool holder response is obtained at two positions on the tool holder, specified as joint coordinates. Next, the tool is inserted into the chuck and the assembly response is measured at the same positions on the tool holder. The free–free response of the tool component is separately computed using FEM modeling at the corresponding joint positions. The density and Young’s modulus of the tool are 14450 Kg/m3 and 5.8!1011 N/m2, respectively. Using the joint model B as above, the RC equations are set up and used in the optimization routine to obtain the joint parameters. Figs. 5 and 6 show the frequency response of spindle and tool at two different positions; one on the tool tip and the other on the tool-holder. In these figures, the major contribution of the joint between two components, i.e. spindle and tool, in the frequency range considered is the introduction of the mode at 1400 Hz. The GA optimization routine proceeds by considering a range of 1 to 1!1012 N/m for stiffness and 1–100 N s/m for damping coefficients for the two joint springs as the feasible range. Fig. 7 shows the convergence of GA to the minimum values after 30 iterations and Table 2, lists the identified joint parameters. Using the parameters in Table 2, tool-spindle assembly response is regenerated using the RC equations. Fig. 8 depicts the RC-based response versus the FRF obtained by conducting modal test on the assembly. It is seen that the RC based response follows the experimental response closely. In particular, the mode at 1400 Hz, which reflects the joint dynamics, is successfully predicted. However, when a joint of type A is used along with the linear FRFs, or when the joint is modeled as rigid (HJZ0), this mode does not appear.
Fig. 8. Predicted and measured frequency response for FADAL milling machine.
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D1
D2 L1
Short tool Medium tool
L2
L1
L2
D1
D2
30 mm 55 mm
30 mm 30 mm
20 mm 25 mm
25 mm 25 mm
Fig. 9. Tool specifications for the experiment on DECKEL milling machine.
Once the joint parameters are identified, they may be used for any other tool-spindle combination, provided that the attachment procedure is similar. Therefore, it may be envisaged that a database of tool FRFs can be created for various standard tool geometries. Then, the spindle FRF is measured only once and is combined with the tool response for quick generation of the assembly response for any toolspindle combination. To further substantiate this statement, a second experiment was carried out. Two cutting tools of different geometries as reported in Fig. 9 were built. For convenience, the tools were built in the form of cylinders, i.e. with no flutes. The medium tool was inserted into the spindle of a 3-axis DEKEL CNC milling machine. The length L2 of the tools was inserted in the spindle, so the joint DOFs were considered along this length. Fig. 10 shows the experimentally obtained FRFs of the spindle and spindlemedium tool assembly. Similar procedures as described above were carried out to obtain the joint parameters for the combination of spindle and medium tool. After extracting the joint parameters using inverse RC method along with GA-based optimization, these parameters were used for coupling of a shorter tool with the spindle. Fig. 11 shows the short tool-spindle assembly FRF obtained from RC theory in comparison with the FRF obtained directly from experiment. It may be seen that the RC method has predicted the response of the assembly with reasonable accuracy.
Fig. 11. Measured spindle and short Tool-spindle assembly response compared with predicted response.
6. Conclusions Receptance coupling method can be used to predict response of tool-spindle combinations on the machine tool spindle. In RC method, the accuracy of predicted results is dependent on having rotational degrees of freedom in RC equations. Since RDOF responses are not easily available from experiment, a special joint model is presented here which can simulate the rotational stiffness of the joint and thus, can predict the correct assembly displacement response, without the need to include rotational FRFs in RC equation. An optimization method based on Genetic Algorithm is used to extract joint parameters. This method is robust because it rarely converges to local minima and is not sensitive to starting point. Also, in our experience, if the frequency range is chosen such that the joint parameters affect the response in that range, the algorithm is always able to find good values for the parameters. Comparison between predicted and measured results in Fig. 8 validates the joint model and the identified joint parameters. Joint parameters identified using one set of tool spindle assembly measurement can be used for generating the assembly response for any tool-spindle combinations attached under similar conditions.
Acknowledgements
Fig. 10. Spindle and medium Tool-spindle assembly response.
The first experiment reported here was conducted during the presence of the first author in the manufacturing automation lab, department of mechanical engineering, the University of British Columbia, Canada. The first author wishes to acknowledge the assistance and helpful suggestions offered by Prof Y. Altintas. This research is also partially supported by grant funds from Sharif University of Technology, whose contribution is gratefully acknowledged.
M.R. Movahhedy, J.M. Gerami / International Journal of Machine Tools & Manufacture 46 (2006) 243–251
References [1] Y. Altintas, E.M. Budak, Analytical prediction of stability lobes in milling, Annals of CIRP 44 (1995) 357–362. [2] Y. Altintas, Analytical prediction of three dimensional chatter stability in milling, Proceeding of Japan Society of Mechanical Engineers 44 (2001) 717–723. [3] T.L. Schmitz, R. Donaldson, Prediction of high speed machining dynamics by substructure analysis, Annals of CIRP 49 (2000) 304–308. [4] M.L.M. Duarte, D.J. Ewins, Rotational degrees of freedom for structural coupling analysis via finite deference technique with a residual compensation, Journal of Mechanical Systems and Signal Processing 14 (2) (2000) 205–227.
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[5] A.P.V. Urgueira, Dynamic analysis of coupled structures using experimental data, PhD Thesis, Imperial College of London, 1989. [6] S.S. Park, Y. Altintas, M. Movahhedy, Receptance coupling for end mills, International Journal of Machine Tool and Manufacture 43 (2003) 889–896. [7] J.H. Wang, C.M. Liou, Experimental identification of mechanical joint parameters, Journal of Engineering for Industry 113 (1991). [8] Y. Ren, C.F. Beards, An iterative FRF joint identification technique, Journal of Sound and Vibration 186 (1995) 567–577. [9] D.E. Goldberg, Algorithm in Search, Optimization and Machine Learning, Addison-Wesley, Boston, MA, 1989.
Prediction of spindle dynamics in milling by sub-structure coupling Mohammad R. Movahhedy*, Javad M. Gerami Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Azadi Ave., Tehran, Iran Received 24 February 2005; accepted 26 May 2005 Available online 8 August 2005
Abstract The Stability of machining process depends on the dynamics of the machine tool, among other things. However, the dynamics of the machine tool changes when the tool is changed. To avoid the need for repeating the measurements, sub-structuring analysis may be used to couple the tool and spindle frequency response functions. A major difficulty in this approach is the determination of joint stiffness and damping between the two sub-structures. In particular, the measurement of rotational responses (RDOFs) at joints is a difficult task. In this research, a simple joint model that accounts for RDOFs is proposed. It is shown that this model avoids RDOF measurement while taking into account the bending modes. An optimization method based on genetic algorithm is employed to find parameters of the joint model. Receptance coupling analysis is used to couple the machine tool and tool FRFs obtained from experiment and FE, respectively. The RC based response obtained in this way is compared with experimental FRF which shows good agreement and confirms that the joint model has been successful in predicting the tool bending modes. q 2005 Elsevier Ltd. All rights reserved. Keywords: Sub-structure coupling; Machining stability; Chatter vibration; Genetic algorithm
1. Introduction The rising trend in the use of high speed machining in aerospace, automotive and other industries has presented new challenges for the machine tool manufacturers and users. High speed machining results in higher material removal rates and better surface quality, but higher machine tool strength and higher precision are required. In particular, the dynamics of the machine tool has a determining effect on the precision and removal rate of the machining process due to the possibility of occurrence of self-excited chatter vibration. For example, in machining of turbo jet engines, long and flexible tools are used which are prone to the chatter occurrence. If cutting parameters such as depth of cut and cutting speed are not selected appropriately, machining process can become unstable resulting in serious damage to machine tool spindle and scrapping of expensive workpieces. It is well established that the occurrence of chatter is dependent on three factors: cutting conditions, work piece * Corresponding author. Tel.: C98 21 616 5505; fax: C98 21 600 0021. E-mail address: [email protected] (M.R. Movahhedy).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.05.026
material properties and the dynamics of machine tool system. The stability lobe diagram, which predicts stable sets of machining parameters, is commonly used to select best parameters for avoiding chatter. Knowledge of the machine tool dynamics, i.e. its frequency response (FRF), at the tool tip is an essential requirement for obtaining stability lobe diagrams [1,2]. Such system responses are traditionally obtained by performing modal tests on the machine tool or by numerical modeling. Experimental FRF is the most reliable method for systems with complex geometries, but is expensive and time consuming, particularly if it has to be repeated for every setup. Analytical and numerical FRF, on the other hand, will inevitably contain approximations and should be carefully validated if system is complex and comprised of many parts assembled together. However, it may be used more reliably for simpler, e.g. single piece, systems. In a milling machine, the response of the machine at the tool tip is mostly affected by the dynamics of spindle, supported on its bearings, and the cutting tool, which is usually the most flexible part of the system. The cutting tool is usually attached to the spindle through a standard tool holder. Due to complexity of spindle geometry and the effects of bearing stiffness and damping, it is customary that the response of the combined spindle-tool holder-tool
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system is obtained from modal testing. However, when a new tool is attached to the spindle, the dynamics of the system changes, and this change may be significant from one setup to another, because the tool is the most flexible part of the assembly and is more prone to vibration. As a result, the modal test may have to be repeated for every new tool setup. However, since it is only the tool that changes in the system, it is possible to obtain the dynamic response of the spindle system—without the tool—only once, and then combine it with the response of the tool by coupling of the two substructures. The response of various tools may be generated either experimentally or by numerical modeling. Due to relative simplicity of the cutting tool structure, it is expected that numerical modeling can yield reliable frequency response functions (FRF) for various cutting tools. Tool holder may be considered as part of spindle, tool, or as a separate substructure. In structural dynamics, the coupling of responses of two separate systems to determine the response of their assembly is known as receptance coupling method (RC). The RC method can significantly reduce the cost of obtaining system response for various setups. However, a major problem in using this method is in identification of stiffness and damping properties introduced to the assembly at joints between substructures. For example, Schmitt et al. [3] combined the analytical tool response and coupled it with spindle FRF, but they adjusted joint parameters in a way that the results of analysis match those of the experiment. Another obstacle in using the RC method is in measuring of responses due to rotational degrees of freedom (RDOF). In the structures which carry bending loads, RDOFs have a significant effect on the dynamics of the structure [4]. However, measurement of RDOF responses is technically difficult. Common methods use a finite difference scheme to obtain the RDOF responses from responses of two linear accelerometers. New laser instruments for angle measurement are recently introduced, but are usually expensive and may not cover all the required data. However, the inclusion of RDOF responses is essential for successful coupling of substructure FRFs [5]. While the rotational FRFs for the tool may be readily generated numerically, the corresponding experimental FRFs are not available for the spindle system. This research proposes a joint model which avoids RDOF measurements for prediction of machine tool assembly FRF. In this model, two linear joint elements are used and the applicability of the model in successful coupling is numerically verified. Using this model, the spindle-tool assembly FRF may be obtained without using rotational FRFs. The model is then applied to real machine tool cases and an optimization method based on genetic algorithm is used to identify joint parameters between tool and spindle. The results are verified through comparison with experimental results.
2. Receptance coupling theory The dynamics of a complex structure can be obtained by combining the dynamics of its substructures either in the modal space or in the response space. The former approach is called component mode synthesis and usually involves approximations due to curve fitting errors. Besides, the effects of high frequency modes that are truncated from analysis are not included. The latter approach is called receptance-coupling (RC) method in which, the effects of high frequency modes are inherently present in the data, but the presence of noise in the experimental data may hinder the success of the coupling procedure. In the RC method, the FRFs of the components of a structure are independently obtained, and are then coupled together analytically by applying the compatibility and equilibrium conditions. Since the components are assembled through mechanical joints that may have mass and stiffness, the dynamics of the joint should be included in the assembled system. If the joint properties are known, this amounts to a direct problem of combining the data [6]. However, usually the joint properties are not known and cannot be measured independently. In such cases, an inverse problem should be solved in which the joint properties are identified from the responses of components and assembly. This is called joint identification problem. Let two substructures with given frequency responses h and h 0 , are coupled together to form a structure whose response is denoted by g. Also, let the degrees of freedom of the substructures are divided into those which are at the joint and those away from the joint, and denote these by subscripts j and n, respectively, and reorder the DOFs such that similar DOFs are grouped together. For the two components, the dynamics equation can be written as #( ) " fn hnn hnj xn Z ; xj hjn hjj fj C F J (1) #( ) " 0 0 fn 0 hn n hn 0 j 0 xn 0 Z 0 xj 0 hj 0 n 0 hj 0 j 0 fj 0 C F J In this equation, for example, hnj shows the response at a non-joint dof n due to force at a joint dof j. For the assembled structure, we have: 8 9 2 38 f 9 n > > 0 0 g g g g x > > > nn nj nj nn n > > > > > > > > > = <
Equilibrium condition requires that F j Z KF j
0
(3)
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and compatibility condition requires that Xj0 K Xj Z H j F j
(4)
where Hj is the receptance matrix of joint, defined as H J Z ½K C juCK1
(5)
in which K and C are stiffness and damping matrices of structural joints. Each element of the FRF matrix in Eq. (2) is an individual frequency response function for corresponding excitation and response positions. We can obtain these FRFs by substitution of Eqs. (1) and (3) in Eq. (4). For example, a non-joint DOF of structure is obtained as a function of substructures’ and joint FRFs as: gnn Z hnn K hnj ðH J C hjj C hj0 j 0 ÞK1 hjn
2
where Annff means linear response due to linear excitation f, Annfm means linear response due to rotational excitation m, and Annmf means rotation due to linear excitation f, etc. [6].
3. Tool-spindle joint model In this section, we will discuss the issue of coupling the FRF of a cutting tool with the FRF of spindle on a milling machine. It is assumed that the dynamics of the machine mainly contributed by its spindle and the other parts of its structure are relatively rigid. Furthermore, the tool holder is considered as part of the spindle assembly for convenience. Due to the complex geometry of the spindle, and since it does not change considerably over time, it is preferred to obtain the response of this substructure by modal testing. The tool, on the other hand, has a much simpler geometry and its response may be obtained analytically or numerically, e.g. from an FE analysis. The difficulty here is that for the spindle, only the first element of the matrices given in Eq. (7) is readily available from modal test. For tool, these elements are readily available from FE analysis. It is shown [6] that if rotational DOFs are not included in the RC equations, the coupling exercise is likely to fail. Another important issue in RC is proper modeling of the tool-spindle joint. The joint used for connection between the two carries both a force and a moment. To model the flexibility of such a joint, a combination of a linear spring and a rotational spring may be used. Here again, the rotational DOFs are involved. Here, instead, a couple of parallel linear springs are used which together can carry both a force and a moment. To verify the equivalence of these two models, a numerical exercise using FE analysis is carried out. In this exercise, a spindle-tool system is
3
3 2
(6)
If rotational DOFs are taken into account, each of the FRFs in Eq. (6) is indeed a matrix composed of four response functions such as " # Annff Annfm ½Ann Z ; (7) Annmf Annmm
5
4
Model A
Model B
1
1
Fig. 1. Model A with rotational and linear joint elements versus Model B with two linear joint elements.
considered in which the joint between the two components is modeled in two different ways; model A connects the two components by a linear and a rotational spring, while model B connects the two through a couple of parallel linear springs (Fig. 1). Stiffness and damping parameters of the two sets of springs are calculated in a way that both joint models introduce similar modes to the assembled system. Table 1 shows the first four frequencies of the assembled systems for both models. An examination of the corresponding mode shapes shows that the second mode at 315 Hz is the first tool mode, the fourth mode at 1716 Hz is the first spindle mode, and the first and third modes at 194 Hz and 948 Hz are the contributions of the joint springs. The mode at 194 Hz represents the rotational motion at the joint, while the mode at 948 Hz represents linear displacement at the joints. Fig. 2 shows the two joint mode shapes for both models exhibiting equivalent shapes. Next, the RC method (Eq. 6) is used to calculate the response of the assembled system from the component FRFs for the two joint models. For model A, all the linear and rotational DOFs of the components are included in RC equations, while for model B, only linear DOFs are considered. Fig. 3 shows that the FRFs of the two models are similar and exhibit all the modes. However, if only linear DOFs are considered for model A, it will not exhibit the mode at 194 Hz that is related to the joint rotational stiffness. This is shown in Fig. 4, where such response is compared with the assembly response computed directly Table 1 The first four natural frequencies of tool-spindle assembly Natural frequencies in Hz Model A Model B
194.25 194.22
314.99 315.30
947.92 947.95
1716.1 1715.8
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Fig. 2. First and third tool-spindle assembly mode shapes at 194 and 947 Hz.
from FE analysis. On the other hand, model B exhibits the first mode because of the moment effect of the two linear springs. From this simple exercise it is concluded that when the linear assembly response at the tool tip is required, e.g. for stability analysis of the machining process, the exclusion of rotational DOFs may result in the disappearance of some modes with significant effects on the stability analysis. However, using model B for the joint compensates this shortcoming and removes the need for measurement of the rotational responses of spindle sub-structure. Finally, it should be mentioned that the assembly response is very sensitive to the joint parameters, and a change in these parameters will considerably shift the peaks affected by joint dynamics.
this practice will involve inverting of matrices with very small elements and low ranks, and is very sensitive to noise and errors that are inherent to the experimental data. For instance, if an error of only a few percent is introduced to the data, the identified joint properties may show much larger difference, e.g. 50%, from the actual levels. Researchers have presented procedures that avoid matrix inversion as much as possible to reduce numerical errors (see e.g. [7,8]).
4. Joint identification In this section, based on the joint model introduced and verified above, the RC method is used for coupling of the tool and spindle system. The first step for this purpose is to obtain the joint properties. If in Eq. (6) or similar equations derived from Eq. (2), the assembly and the components’ FRFs are known, the joint receptance matrix (Hj) can be obtained by algebraically inverting the equations. However,
Fig. 3. Predicted response of models A and B by RC analysis.
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Fig. 4. The effect of elimination of RDOF response in FRF prediction of Model A.
Alternatively, Eq. 6 can be solved as a nonlinear optimization problem for the joint properties. The optimization method finds the appropriate joint parameters that minimize the difference between RC-based assembly FRF and the corresponding experimental FRF. The objective function may be constructed as the following vector: ( ) Reðgp K gm Þ jgj Z (8) Imðgp K gm Þ in which gp and gm are predicted (from Eq. (6)) and measured receptances of the assembly. Euclidian norm of vector g is used as the objective function. Since the response values are complex, it is divided into real and imaginary parts. Also response values may be predicted and measured at various joint and non-joint DOFs and at a vast range of frequency points to expand the data space. In this work, the genetic algorithm (GA) search method is used to minimize the objective function. GA is a global search method which acts on the space of results [9]. All variables are coded with binary strings. A population of acceptable values is generated on the space of results. Each individual in the population is a binary string that can be decoded into real space. Individuals are evaluated with respect to their objective function values. Best individuals which minimize objective function are selected to generate a new population. Next population are generated by cross over, mutation and reinsertion of selected individuals. New individuals are evaluated and the best of them are selected to generate new population and the loop is continued to minimize objective function. The GA is preferred over other classic optimization algorithms because the latter
algorithms are generally very sensitive to the starting point, and quite often converge to local minimum and stop prematurely, while GA is a global search method that rarely converges to local minima. In addition, it is not necessary to specify a start point for GA; the space of acceptable results is enough for GA to find the minimum of a function. The parameters that are binary coded for the joint identification problem are the joint parameters, i.e. the stiffness and damping coefficients at the joints. The initial population consists of member sets of randomly generated coefficients within the specified range. For every member of this population, the objective function is calculated and evaluated. Next, 10% of the population that have yielded highest function values are erased and the rest of the population are crossed over at a probability of 0.7. In other words, 70% of the new population is generated by dividing the binary strings into parts and putting various parts together. Furthermore, 5% of the population is changed by mutation, in which a digit in the target string is switched from 0 to 1 or vice versa. Some of the member sets from the old population are also included. The new combinations of joint parameters formed by such crossover, mutation and reinsertion forms the new population. The function values are calculated for this new population and the process is repeated until it converges to the minimum solution. Evaluation of convergence rate is carried out at intervals and if the solutions are not satisfactory, the process is started over with a new set of population members. The run time of the algorithm depends on the number of population members, the string length and number of iterations. Increasing the number of members
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Fig. 5. Frequency response of spindle and tool-spindle assembly at the tool tip.
reduces the risk of convergence to a local minimum. In most problems, this number is set between 30 and 100. A longer string results in a more accurate solution. Also, increasing the number of members reduces the number of iterations in general.
In this work, the genetic algorithm module of MATLAB is used for finding joint parameters that minimize Eq. (8). The time to convergence depends on number of equations involved and the frequency range of interest. In the problem considered here, at least four RC equations were written for
Fig. 6. Frequency response of spindle and tool-spindle assembly on the tool-holder.
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Fig. 7. GA convergence to minimum solution.
Table 2 identified joint parameters k1 (N/m)
k2 (N/m)
c1 (N s/m)
c2 (N s/m)
6.31!108
9.82!1011
7.86
6.13
each frequency point in the specified range, and the norm of error at all frequencies was calculated as the objective function. Typically, the algorithm converged to the optimum solution within around 30 iterations, and in about 10–20 min on a Pentium IV processor.
5. Applications and verification To apply the above procedures, two experimental tests have been carried out. In the first experiment, a vertical 3-axis milling machine and a 19.05 mm diameter end
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mill were used. In this test, first, a mechanical chuck is attached to the spindle, without the tool and the spindletool holder response is obtained at two positions on the tool holder, specified as joint coordinates. Next, the tool is inserted into the chuck and the assembly response is measured at the same positions on the tool holder. The free–free response of the tool component is separately computed using FEM modeling at the corresponding joint positions. The density and Young’s modulus of the tool are 14450 Kg/m3 and 5.8!1011 N/m2, respectively. Using the joint model B as above, the RC equations are set up and used in the optimization routine to obtain the joint parameters. Figs. 5 and 6 show the frequency response of spindle and tool at two different positions; one on the tool tip and the other on the tool-holder. In these figures, the major contribution of the joint between two components, i.e. spindle and tool, in the frequency range considered is the introduction of the mode at 1400 Hz. The GA optimization routine proceeds by considering a range of 1 to 1!1012 N/m for stiffness and 1–100 N s/m for damping coefficients for the two joint springs as the feasible range. Fig. 7 shows the convergence of GA to the minimum values after 30 iterations and Table 2, lists the identified joint parameters. Using the parameters in Table 2, tool-spindle assembly response is regenerated using the RC equations. Fig. 8 depicts the RC-based response versus the FRF obtained by conducting modal test on the assembly. It is seen that the RC based response follows the experimental response closely. In particular, the mode at 1400 Hz, which reflects the joint dynamics, is successfully predicted. However, when a joint of type A is used along with the linear FRFs, or when the joint is modeled as rigid (HJZ0), this mode does not appear.
Fig. 8. Predicted and measured frequency response for FADAL milling machine.
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D1
D2 L1
Short tool Medium tool
L2
L1
L2
D1
D2
30 mm 55 mm
30 mm 30 mm
20 mm 25 mm
25 mm 25 mm
Fig. 9. Tool specifications for the experiment on DECKEL milling machine.
Once the joint parameters are identified, they may be used for any other tool-spindle combination, provided that the attachment procedure is similar. Therefore, it may be envisaged that a database of tool FRFs can be created for various standard tool geometries. Then, the spindle FRF is measured only once and is combined with the tool response for quick generation of the assembly response for any toolspindle combination. To further substantiate this statement, a second experiment was carried out. Two cutting tools of different geometries as reported in Fig. 9 were built. For convenience, the tools were built in the form of cylinders, i.e. with no flutes. The medium tool was inserted into the spindle of a 3-axis DEKEL CNC milling machine. The length L2 of the tools was inserted in the spindle, so the joint DOFs were considered along this length. Fig. 10 shows the experimentally obtained FRFs of the spindle and spindlemedium tool assembly. Similar procedures as described above were carried out to obtain the joint parameters for the combination of spindle and medium tool. After extracting the joint parameters using inverse RC method along with GA-based optimization, these parameters were used for coupling of a shorter tool with the spindle. Fig. 11 shows the short tool-spindle assembly FRF obtained from RC theory in comparison with the FRF obtained directly from experiment. It may be seen that the RC method has predicted the response of the assembly with reasonable accuracy.
Fig. 11. Measured spindle and short Tool-spindle assembly response compared with predicted response.
6. Conclusions Receptance coupling method can be used to predict response of tool-spindle combinations on the machine tool spindle. In RC method, the accuracy of predicted results is dependent on having rotational degrees of freedom in RC equations. Since RDOF responses are not easily available from experiment, a special joint model is presented here which can simulate the rotational stiffness of the joint and thus, can predict the correct assembly displacement response, without the need to include rotational FRFs in RC equation. An optimization method based on Genetic Algorithm is used to extract joint parameters. This method is robust because it rarely converges to local minima and is not sensitive to starting point. Also, in our experience, if the frequency range is chosen such that the joint parameters affect the response in that range, the algorithm is always able to find good values for the parameters. Comparison between predicted and measured results in Fig. 8 validates the joint model and the identified joint parameters. Joint parameters identified using one set of tool spindle assembly measurement can be used for generating the assembly response for any tool-spindle combinations attached under similar conditions.
Acknowledgements
Fig. 10. Spindle and medium Tool-spindle assembly response.
The first experiment reported here was conducted during the presence of the first author in the manufacturing automation lab, department of mechanical engineering, the University of British Columbia, Canada. The first author wishes to acknowledge the assistance and helpful suggestions offered by Prof Y. Altintas. This research is also partially supported by grant funds from Sharif University of Technology, whose contribution is gratefully acknowledged.
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