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Extending the inverse receptance coupling method for prediction of tool-holder joint dynamics in milling
Journal of Manufacturing Processes 14 (2012) 199–207
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Technical paper
Extending the inverse receptance coupling method for prediction of tool-holder joint dynamics in milling Mohammad Mahdi Rezaei ∗ , Mohammad R. Movahhedy, Hamed Moradi, Mohammad T. Ahmadian Center of Excellence in Design, Robotics and Automation (CEDRA), Department of Mechanical Engineering, Sharif University of Technology, Azadi Ave, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 4 April 2011 Received in revised form 3 October 2011 Accepted 21 November 2011 Available online 20 December 2011 Keywords: Inverse receptance coupling Joint model Tool-holder FRF Milling dynamics
a b s t r a c t Recently, receptance coupling substructure analysis (RCSA) is used for stability prediction of machine tools through its dynamic response determination. A major challenge is the proper modelling of the substructures joints and determination of their parameters. In this paper, a new approach for predicting tool tip FRF is presented. First, inverse RCSA formulation is extended so that the holder FRFs can be identified directly through experimental modal tests. The great advantage of this formulation is its implementation in arbitrary point numbers along joint length. Therefore, in comparison with previous inverse RCSA approaches, a more realistic joint model can be considered. In addition, due to applying the new approach, additional costly modal tests on the gauged tool are not required. This characteristic makes it possible to determine the holder FRFs without separating the tool; especially in situations where the holder end is inaccessible. The inclusion of joint parameters effect in the identified holder FRFs is another main advantage of such approach. Consequently, for identification of joint parameters, there is no need to use common error optimization based on fitting methods. The effect of overhang length is investigated through some analytical study and also experimental validation. Results show that the predicted tool tip FRF is exact in analytical case. Moreover, due to less noise effect, the predictions based on identified FRFs of longer tools are more accurate than the shorter ones (in experimental case). © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction Prediction and prevention of unstable chatter vibration is essential for achieving high quality and efficiency of machining processes. In the majority of methods used to find the stable free chatter regions of machining, finding the tool tip FRF is essential. End milling is one of the machining processes with most prone to the chatter. Budak and Altintas [1] presented a method to determine the stability lobe diagram in milling using tool tip FRF. In practice, experimental modal analysis is used to determine the tool tip FRF. However, using this method is time consuming and cumbersome, because the tool response changes as the tool is changed. In addition, for the cases where the experimental modal analysis is not possible such as flexible tools or micro milling, modelling is the only appropriate alternative. On the other hand, the modelling method based on only analytical results is a difficult task and often unreliable, especially for complex systems such as milling machine. An alternative is to couple the analytical response of the tool with the experimental response of the machine.
∗ Corresponding author. Tel.: +98 21 66165511; fax: +98 21 66000021. E-mail address: rezaei [email protected] (M.M. Rezaei).
The concept of receptance coupling has been extensively used for various applications. Wang and Liou [2] and Tsai and Chou [3] obtained the FRFs of a coupled system consisting of two planes from their separate FRFs. Ren and Beards [4] presented a more convenient approach of receptance coupling for application in linear joint modelling. In recent year, many works have been devoted to apply the RCSA in prediction of tool tip FRF in machining. Schmitz et al. [5] used RCSA to predict the FRFs of tool-spindle system. In their work, FRFs of the tool was found by analytical methods and was coupled with the experimental FRF of tool holder tip to find the assembly FRF. In this way it is possible to find the FRF of the assembled system when the tool is changed, without repetition of the experimental modal analysis. The main obstacle of this method is the joints modelling. Schmitz et al. [5] used a translational spring/damper to model the joint and identified its value by fitting the results of the experimental test and modelling. Liu and Ewins [6] and Park et al. [7] showed that neglecting the rotational FRFs through coupling process produces a large error in (especially in predicting FRFs of long length tools [7]). However, measuring rotational FRFs is difficult and expensive. To extract the rotational FRFs of tool holder, they proposed the inverse RCSA method. In this method, the direct FRF of holder set is determined by experimental modal test on gauged tool that has the same joint condition as the sample tool. Also, the tool rotational FRFs are identified through
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some experimental modal test on the tool connected to holder. Park and Chae [8] used this method for FRF prediction in modular tool applying receptance coupling method. Schmitz and Duncan [9] presented a three-component model including spindle, tool and tool holder and used the translational and rotational springs/dampers between the components. They found the values of the spring/damper by fitting the results of experimental test. Burns and Schmitz [10] investigated an analytical method to study the effect of tool overhang length on magnitude of joint parameters. They showed that the trend of joint parameters may be predictable only when a full connecting matrix is considered for it. Kivanc and Budak [11] considered complex geometry of fluted tool by a cylinder of equivalent diameter and used previous coupling method to predict tool-tip FRF. In the majority of mentioned works, the portion of tool located inside the tool holder is assumed as part of the holder while the joint is located between the overhang and tool holder substructure. If a multiple point model along the joint is applied, the prediction accuracy is increased. Schmitz and Duncan [12] developed the receptance coupling through multi point joint model. In other works, the accuracy of predictions was improved by modifying physics of the model through considering the joint along the contact interface. Movahhedy and Gerami [13] presented a model based on two spring/damper set at the joint ends. Consequently, it is not required to measure the rotational FRFs of the tool holder while the rotational modes are considered. In their work, a genetic algorithm (GA) was used to determine the joint stiffness and damper. A more complete model based on a continuous profile of the spring/damper along the tool holder interface was developed by Ahmadi and Ahmadian [14]. Erturk et al. [15–17] used a spring/damper set at the joint ends using the average values of the joint parameters provided in the literature. They demonstrated the capability of their model in optimizing the characteristics of the milling machine components. Namazi et al. [18] presented a joint model that included a set of translational and rotational springs/dampers along the joint. They found that the joint parameters extracted from a short length tool do not yield good results for prediction of the FRF of the longer overhang. Many of the RCSA works use fitting to experimental data as a way to extract joint parameters. However, this approach is not universally successful, because the search region for joint parameters is very wide and the process is time consuming. In addition, the results of the receptance coupling are very sensitive to joint parameters, and these values are not very reliable. Schmitz et al. [19] used a method based on contact model in commercial finite element software to determine joint parameters. In that method, simplifications and averaging was used to convert a complex 3D-model to a simple model of beam element. In their approach, successive averaging stages were performed which caused accuracy reduction. In this paper, a new method is proposed for direct determination of the tool tip FRF without identification of joint parameters by fitting method. In the first stage, the FRF of whole system (including tool and holder set) at tool tip is formulated as function of its substructures FRFs. Then, the inverse receptance coupling is extended for arbitrary point numbers in joint modelling. Using the extended procedure, the tool/holder set FRFs is obtained by subtraction of the analytical tool FRFs from whole assembly FRFs. The resulting FRFs represent the dynamics of both the tool holder and the joint. So, the whole assembly FRF at the tool tip with various overhang length is obtained through coupling of the extracted tool holder FRFs with analytical tool FRFs. In contrast to the previous works, using this method makes it possible to extract tool holder FRFs; especially when holder is inaccessible for direct measurements; e.g. at the joint surface inside the spindle bore. This paper is set as follows. In Section 2, the formulation of the extended RCSA and its inverse and their application in extracting
the tool holder FRFs is described. In Section 3, the accuracy and efficiency of the proposed approach is analytically investigated. In Section 4, experimental results are provided to evaluate the accuracy of the proposed method for various tool overhang lengths. Finally, conclusions are provided in Section 5. 2. The receptance coupling substructure analysis method and its inverse 2.1. Receptance coupling substructure analysis (RCSA) A simplified model of a milling machine is depicted in Fig. 1. In this model, the structure is constituted of two components: the tool (part B), and the tool holder and other parts of the structure (part A). The coupling process is performed through n points along tool holder/tool joint. Considering the FRFs of part A at n points, its n × n FRF matrix is written as:
⎧ ⎫ XA,1 ⎪ ⎪ ⎪ ⎨ XA,2 ⎪ ⎬
⎡
HA,11 ⎢ HA,21 =⎢ . .. ⎣ .. ⎪ ⎪ . ⎪ ⎪ ⎩ ⎭ XA,n HA,n1
HA,12 HA,22 .. . HA,n2
⎤⎧
··· ··· .. . ···
⎫
HA,1n ⎪ FA,1 ⎪ ⎪ ⎪ HA,2n ⎥ ⎨ FA,2 ⎬ ⎥ .. ⎦ .. ⎪ ⎪ . ⎪ ⎩ . ⎪ ⎭ FA,n HA,nn
(1)
where X and F denote the displacement and corresponding force vectors at each point, respectively. For instance, considering two degrees of freedom of lateral displacement and rotation degree for each point, the corresponding force vectors will be lateral force f and moment M. HA,ij is the FRF sub-matrix between points i and j of part A. For instance, the force vector and FRF sub matrix corresponding to displacements of point 1 due to applying a force at point 2 are:
XA,1 =
vA,1
,
A,1
HA,12 =
hA12,ff hA12,Mf
hA12,fM hA12,MM
,
FA,2 =
fA,2 MA,2
(2) Considering tool tip FRF at m point, the FRF matrix of part B and whole structure T will become:
⎧ ⎪ ⎨
XB,1 XB,2 . . ⎪ ⎩ . XB,n+m
⎫ ⎡ ⎪ ⎬ ⎢ =⎣ ⎪ ⎭
⎧ ⎫ XT,1 ⎪ ⎪ ⎪ ⎨ XT,2 ⎪ ⎬
⎡
HB,11 HB,21 . . .
HB,12 HB,22 . . .
HB,(n+m)1
HB,(n+m)2
HT,11
⎢ HT,21 =⎢ . .. ⎪ ⎪ ⎣ .. ⎪ ⎩ . ⎪ ⎭ XT,m
HT,m1
HT,12 HT,22 .. . HT,m2
··· ··· . . . ···
HB,1(n+n) HB,2(n+m) . . . HB,(n+m)(n+m)
⎤⎧ ⎪ ⎨ ⎥ ⎦ ⎪ ⎩
FB,1 FB,2 . . . FB,(n+m)
⎫ ⎪ ⎬ ⎪ ⎭
⎫ ⎤⎧ · · · HT,1m ⎪ FT,1 ⎪ ⎪ ⎪ · · · HT,2m ⎥ ⎨ FT,2 ⎬ ⎥ .. .. ⎦ .. ⎪ ⎪ . . ⎪ ⎩ . ⎪ ⎭ ···
HT,mm
(3)
(4)
FT,m
On the other hand, the FRF matrix of the whole structure can be obtained by coupling its substructure at n points along the joint through RCSA method. To simplify the formulation, points are divided to contact points with subscript C¯ and non-contact points ¯ so Eqs. (2), (3) and (4) are rewritten as: with subscript n; XA,¯c = SA,¯cc¯ FA,¯c
XB,¯c XB,n¯
=
SB,¯cc¯ SB,n¯ ¯c
SB,¯cn¯ SB,n¯ n¯
FB,¯c FB,n¯
XT,n¯ = ST,n¯ n¯ FT,n¯
(5) (6) (7)
At the points where RCSA is performed and at points on the tool, equilibrium and compatibility equations are described as follows:
FA,¯c + FB,¯c = 0 , XA,¯c = XB,¯c
FB,n¯ = FT,n¯ XB,n¯ = XT,n¯
(8)
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Fig. 1. Two part model of the milling machine including tool and tool holder.
Substituting the values of XA,¯c and XB,¯c from Eqs. (5) and (6) into Eq. (8) yields: −SA,¯cc¯ FB,¯c = SB,¯cc¯ FB,¯c + SB,¯cn¯ FT,n¯ ⇒ FB,¯c = −(SA,¯cc¯ + SB,¯cc¯ )−1 SB,¯cn¯ FT,n¯ (9) Substituting FB,¯c in Eq. (6), XT,n¯ at the end points of the tool is obtained as: −1 XT,n¯ = (SB,n¯ n¯ − SB,n¯ ¯ c (SA,¯c c¯ + SB,¯c c¯ ) SB,¯c n¯ )m×m FT,n¯
(10)
Therefore, according to Eq. (7) the obtained whole system FRFs in terms of its substructure FRFs is given as: −1 ST,n¯ n¯ = SB,n¯ n¯ − SB,n¯ ¯ c (SA,¯c c¯ + SB,¯c c¯ ) SB,¯c n¯
(11)
2.2. Extended inverse receptance coupling substructure analysis In this section, the inverse approach of RCSA is described by presenting the reverse of the above RCSA approach. If all substructure FRFs of a structure are known, the FRF matrix of the assembled structure can be determined through applying RCSA method. This requires that the joint parameters at the interface be known. To avoid determination of joint parameters by fitting methods, an alternative simple method is used. In this method, the tool holder FRFs are determined by subtracting of the analytically obtained tool FRFs from the FRFs of tool holder/tool assembly. The extracted FRFs of tool holder are different from those obtained by performing direct experimental modal test on tool holder, because the former includes the dynamics of the joint parameters between tool holder/tool. Then, this FRF can be directly used to predict assembly FRF for every combination of the same tool holder and any other tool, provided that the joint has the same condition. The details of the developed inverse RCSA method for extraction of the tool holder FRFs at n points along joint length by having the whole structure FRFs at m points on the tool (Fig. 1) is given below. Considering two degrees of freedom for each point of model, the FRF matrix of whole structure T is 2m × 2m. As shown in Fig. 1, n points along the joint are considered for coupling purpose; so the extracted FRF matrix A is a 2n × 2n matrix. Eq. (11) is used to determine the tool holder FRFs. In this equation the FRF matrices of SB,n¯ n¯ , SB,n¯ ¯ c , SB,¯c n¯ and SB,¯c c¯ are the tool FRFs that can be obtained analytically. The FRF matrix of ST,n¯ n¯ is known from experimental modal measurement at tool tip. The aim is to obtain the FRF matrix SA,¯cc¯
by solving Eq. (11). Although the solution of this equation seems to be nonlinear, it is made linear by changing the unknown SA,¯cc¯
to the new unknown matrix H = (SA,¯cc¯ + SB,¯cc¯ )−1 . For determining the FRF matrix at n points of joint interface, the number of measured FRFs of structure, 2m, must be at least equal to the desired number of FRFs, 2n. To simplify the calculation, it is assumed that the same number of measurements m as the number of coupling point n is made. If the measured FRFs of ST,n¯ n¯ are independent, there is a unique series for SA,¯cc¯ . Due to symmetry of the FRF matrices, the number of both independent equations and unknown variables is 2n × (2n + 1)/2. After determination of H, matrix SA,¯cc¯ is simply obtained as follows: H = (SA,¯cc¯ + SB,¯cc¯ )−1 ⇒ SA,¯cc¯ = H −1 − SB,¯cc¯
(12)
As mentioned before, the number of identified FRFs of SA,¯cc¯ is equal to the number of independent measured FRFs of ST,n¯ n¯ . The maximum number of independent measured FRFs is three. This is because, if the FRF sub-matrix at any point is known, the FRFs of tool at any other points can be obtained by applying RCSA method directly. In the considered model, each point has two degrees of freedom and consequently the sub-matrix at any point is 2 × 2; which due to symmetry has only three independent elements. In other words, maximum number of tool holder FRFs is three. Although, this sets a limit for extraction of tool holder FRFs, it also implies that having three FRFs of ST,n¯ n¯ is enough to extract tool holder FRFs so that exact FRFs at any point of structure can be predicted. In this paper, the coupling process is performed by using force-lateral displacement FRFs of two points along the joint length (Fig. 2). In some of the previous works [7,9], connection between two components was at a single point, and the FRFs at the connection point included rotational, translational and cross functions (Hff , Htt , Htf ). However, in following sections, the connection is established at two points (1 and 2) and the concerning FRFs are H11 , H12 , H22 (translational FRFs). It is shown that by considering two connection points, the effect of rotation is automatically included (as also discussed in [13]). The force-lateral displacement FRF matrix of components A, B and whole structure T, Fig. 2, are described as:
vA,1 vA,2
=
hA11,ff hA21,ff
hA12,ff hA22,ff
fA,1 fA,2
(13)
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Fig. 2. Two part model of milling machine used for FRF coupling at two points along the joint.
⎧ ⎫ ⎪ ⎨ vB,1 ⎪ ⎬
⎡
hB11,ff vB,2 ⎢ hB21,ff =⎣ hB31,ff ⎪ ⎪ ⎩ vB,3 ⎭ vB,4 hB41,ff
vT,1 vT,2
=
hT 11,ff hT 21,ff
hB12,ff hB22,ff hB32,ff hB42,ff hT 12,ff hT 22,ff
hB13,ff hB23,ff hB33,ff hB43,ff
fT,1 fT,2
hB14,ff hB24,ff hB34,ff hB44ff
⎫ ⎤⎧ ⎪ ⎨ fB,1 ⎪ ⎬ ⎥ fB,2 ⎦ f ⎪ ⎩ B,3 ⎪ ⎭
(14)
fB,4
(15)
where hAij,ff , hBij,ff and hTij,ff are the force/translational displacement FRFs between points i and j of parts A, B and T, respectively. According to Eq. (11), the FRF matrix of the whole structure T can be obtained through coupling of its substructures A and B. Substituting FRF matrix formulations from Eqs. (13), (14) and (15) into Eq. (11) yields:
hT 11,ff hT 21,ff
hT 12,ff hT 22,ff
=
hB34,ff hB44,ff
hB33,ff hB43,ff
×
˛ ˇ
ˇ
×
+
hB13,ff hB23,ff
hB31,ff hB41,ff hB14,ff hB24,ff
hB32,ff hB42,ff
(16)
where ˛, ˇ and are the elements of the unknown matrix H, given in Eq. (12). Making equal the matrix elements of both sides of Eq. (16), yields the set of equations as follows:
hB13,ff hB31,ff hB13,ff hB32,ff + hB23,ff hB31,ff hB23,ff hB32,ff hB13,ff hB14,ff hB13,ff hB24,ff + hB14,ff hB23,ff hB23,ff hB24,ff hB14,ff hB41,ff hB14,ff hB42,ff + hB24,ff hB41,ff hB24,ff hB42,ff
=
hB33,ff − hT 11,ff hB34,ff − hT 12,ff hB44,ff − hT 22,ff
˛ ˇ
(17)
Using Eq. (17) yields ˛, ˇ and , then the elements of the SA,¯cc¯ matrix are found through Eq. (12). 3. Numerical investigation In this section, numerical modelling is used to investigate the applicability and accuracy of the proposed inverse RCSA approach. The advantage of this approach in contrast to experimental analysis is the elimination of the sources of the experimental error especially associated with the measurement noise. In this investigation, a model of tool holder/tool assembly is considered as shown in
Fig. 3. A continuous distribution of spring/damper is considered to model the connection between tool holder and tool. Such a model has been used in the literature as one of the most precise models that can predict the behaviour of the tool holder/tool connection if the properties of spring and damper are determined properly. In this investigation, we have adopted these properties from [7]. The FRFs of the assembly is obtained through finite element method (FEM). Since the objective of this analysis is to evaluate the effectiveness of the RCSA and IRCSA methods, the Euler–Bernoulli beam elements are used for simplification. However it should be noted that for components with small slenderness ratio, more accurate models such as Timoshenko beam elements can be used. Then, the presented method is used for extraction of the FRFs of the tool holder. In the following, the effect of variation of tool overhang length on the extracted values of the tool holder FRFs (including joint parameters) is studied. Also, the extracted tool holder FRFs for a definite length of tool are used to predict the tool FRF with different overhang lengths. Fig. 4 shows the three tool assemblies with different overhang lengths while other parameters are considered equal (as given in Table 1). For each assembly, tool tip FRF at end points 1 and 2 is found analytically. Next, the tool holder FRFs (including joint parameters) is extracted using the proposed inverse RCSA method. Fig. 5 shows the extracted tool holder FRFs for the three assemblies. It is observed that the extracted FRFs coincide with each other. This confirms this fact that with similar joint conditions, tool overhang length has no effect on extracted holder FRFs. Next, we investigate that the any of the extracted tool holder FRFs can be used to predict the FRFs of other cases. In this investigation, cases A, B and C correspond to the extracted holder FRFs from short, medium and long tools, respectively. Fig. 6 shows the tool-tip FRF obtained by coupling the extracted tool holder FRFs of cases A, B and C for the tool overhang length of case A. Similarly, other extracted FRFs can be used to reconstruct the assembly FRFs for other cases with similar results. It is observed that the predicted FRF results coincide with each other and with the analytically obtained FRFs of the total assembly. Therefore, the predicted holder base FRFs for each tool is independent of the case from which it is extracted. This is because, in numerical analysis, linear modelling is considered for the tool/tool-holder connection. Therefore, increasing tool overhang length is in analogous with increasing in the dynamic loading which does not change the connection FRFs.
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203
Fig. 3. Continuous uniform distribution of spring/damper between tool and holder for analytical modelling of milling machine, tool holder and tool joint.
Fig. 4. Three tool samples with different overhang lengths. Table 1 The nominal specification of the model. Tool Et t Dt Lt1 Lt2 t
Tool holder 540 GPa 14400 kg/m3 20 mm 100 mm 50 mm 0.02
Eh h Lh1 Lh2 Dh1 Dh2 h
Contact parameter 200 GPa 7800 kg/m3 110 mm 70 mm 20 mm 40 mm 0.02
kc c
1.37 × 1012 N/m3 0.35
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Fig. 5. The extracted tool holder FRFs for three cases with different overhang lengths.
Fig. 6. The analytical and predicted tool tip FRF of short tool for three cases of extracted holder FRFs.
In addition, it is shown that the coupling of FRFs at just two points along the interface is sufficient for prediction of the tool tip FRF.
machine and experimental modal analysis is carried out. Also, other joint specifications such as the collet torque are not changed from one sample test to another. The FRFs of each tool at points of 3 and 4 (Fig. 7) are obtained through experimental modal analysis. According the proposed method in Section 2.2, for each of three tools, holder FRFs are extracted by subtracting tool analytical FRFs from the experimental FRFs of the assembled system. Extracted tool holder FRFs for each of three samples are shown in Figs. 8–10. It is observed that the extracted holder FRFs for long and medium length tools are close to each other while that for the short tool contains larger fluctuations. The source of these fluctuations may be attributed to the effect of experimental noise. It should be noted that in the IRCSA method, FRFs of one component are subtracted
4. Experimental investigation In this section, the accuracy and efficiency of the proposed method is investigated through comparing with experimental results. The experimental results are adopted from the previous work [7]. The effect of changing tool overhang length is studied while other parameters are constant. Three carbide cylinder blanks with specifications given in Fig. 7 and density of 14480 kg/m3 and modulus of elasticity of 580 GPa are considered. All samples with joint length of 20 mm are attached in a vertical three axes-milling
Fig. 7. Three blanks of different overhang lengths used for experimental investigation.
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Fig. 8. The direct FRF of point 1 (h11,ff ), extracted from three tools with different overhang lengths.
Fig. 9. The cross FRF of point 1 (h12,ff ), extracted from three tools with different overhang lengths.
Fig. 10. The direct FRF of point 2 (h22,ff ), extracted from three tools with different overhang lengths.
205
Fig. 11. Tool tip FRF of long tool, predicted by extracted tool holder FRFs of different overhang lengths.
from those of total system. So, in the remained parts of signal (with small amplitudes), signal to noise ratio is decreasing. This phenomenon is more adverse around the resonance frequencies due to the existence of computational errors arisen by singularities (in inverse operations). Therefore, as the signal to noise ratio is increased, the final results are more reliable. Such behaviour, that the longer tools with larger signal amplitudes are less prone to noise, is observed in next results. Moreover, the relative distance between the points 3 and 4 (Fig. 7) can be important in some cases. Generally, by decreasing the distance between the measured points, the order of difference between measured signals approaches to that of noise signal. This matter is essentially observed in results of short tool. Under such conditions, to increase the accuracy of results, signal smoothing by using filter methods can be applied (e.g., [9]). Next, extracted holder FRFs for each of the cases short, medium and long cases, shown in Figs. 8, 9 and 10, are used to predict tool tip FRF for different overhang lengths. In these figures, case A, B and C correspond to the extracted holder FRFs of the short, medium and long tools, respectively. Fig. 11 shows the tool tip FRF of the long tool predicted by coupling its analytically obtained FRFs with extracted tool holder FRFs in each of the cases A, B and C. The predicted tool tip FRF have dominant frequencies at 400, 545, 770 and 970 Hz. Fig. 11 shows that the experimentally measured FRFs [7] matches with the predicted FRFs of the long length tool (case C) exactly. This confirms that using presented inverse RCSA method, leads to the desired tool holder FRFs as well. The predicted tool tip FRF of the long tool in the case B (related to extracted holder FRF of medium tool case) is close to the measured FRF. However, in peak frequencies, it predicts larger amplitudes than the measured ones. Again, less agreement is observed for case A, which was expected because its corresponding tool holder FRFs contained large fluctuations too. Fig. 12 shows the medium tool tip FRF for each cases of extracted tool holder FRFs. Similarly, the predicted tool tip FRF for case B coincides perfectly the measured FRF. The tool tip FRF of case C is close to the measured one. This confirms that the extracted holder FRFs for long tool can be used to predict the tool tip FRF of medium tool. For case A, predicted tool tip FRF has larger fluctuations again. Tool tip FRF for short length tool for each cases of extracted tool holder is shown in Fig. 13. It is observed that the predicted tool tip FRF in all cases are close to each other. Therefore it can be concluded that the extracted holder FRFs for the longer tool can be used in prediction of tool tip FRF of shorter tools. But prediction
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Fig. 12. Tool tip FRF of medium tool, predicted by extracted tool holder FRFs of different overhang lengths.
The great advantage of such approach is that joint parameters effect is reflected in the identified holder FRFs. Consequently, unlike the previous works, there is no need for identification of joint parameters through commonly tedious fitting methods. Another advantage of this modified formulation is the extraction of experimental holder set FRFs at arbitrary point numbers along joint length. It is shown that the coupling of FRFs at just two points along the interface is sufficient for prediction of the tool tip FRF. In addition, unlike the previous works using inverse RCSA method, additional modal tests on the gauged tool are not required. Specifically in situations that direct FRF measurement is impossible (when the holder end is inaccessible), using the proposed approach is more applicable. Some analytical investigations and experimental validations are performed to study the effect of overhang length on the accuracy of method predictions. In analytical case, exact predictions of tool tip FRF are obtained regardless of the overhang length of sample tool. According to experimental results, good predictions of assembly FRFs are achieved for sufficiently long tools. However, the results are less accurate for shorter tools, where the experimental noise effect is rather small. In future, this proposed method is developed to be capable of prediction for tool/holder cases with different joint parameters such as tool diameter and material, joint length and contact pressure.
References
Fig. 13. Tool tip FRF of short tool, predicted by extracted tool holder FRFs of different overhang lengths.
of long length tool tip FRF using extracted FRFs of shorter tools is less accurate. This inaccuracy depends partly on the order of the signal to noise ratio in measurements. For shorter tools with larger noise in measurements, less accurate results are obtained. Therefore, if the tools lengths are long enough, predictions are good. 5. Conclusions In this paper, prediction of tool tip FRF is performed through a new extended inverse RCSA method. This convenient approach overcomes the major challenge in the determination of joint parameters in substructures connections. Direct identification of holder FRFs through experimental modal tests are accomplished by extending an inverse RCSA formulation. In this method, the tool/holder set FRFs is obtained by subtraction of the analytical tool FRFs from whole assembly FRFs. Therefore, the tool tip FRF of other tools with different overhang lengths is predicted through coupling of the extracted tool holder FRFs with analytical tool FRFs.
[1] Budak E, Altintas Y. Analytical prediction of chatter stability in milling. Part I: general formulation; Part II: application to common milling systems. Transactions of ASME, Journal of Dynamic Systems, Measurement, and Control 1998;120:22–36. [2] Wang JH, Liou CM. Experimental identification of mechanical joint parameter. Journal of Vibration and Acoustics 1991;113:28–36. [3] Tsai JS, Chou YF. The identification of dynamic characteristics of a single bolt joint. Journal of Sound and Vibration 1998;125(3):487–502. [4] Ren Y, Beards CF. On substructure synthesis with FRF data. Journal of Sound and Vibration 1995;185(5):845–66. [5] Schmitz TL, Davies MA, Kennedy MD. Tool point frequency response prediction for high-speed machining by RCSA. Transactions of the ASME, Journal of Manufacturing Science and Engineering 2001;123:700–7. [6] Liu W, Ewins D. Importance assessment of RDOF in FRF coupling analysis. In: Proceedings of the international modal analysis conference (IMAC), vol. 2. 1999. p. 1481–7. [7] Park SS, Altintas Y, Movahhedy M. Receptance coupling for end mills. International Journal of Machine Tools & Manufacture 2003;43:889–96. [8] Park SS, Chae J. Joint identification of modular tools using a novel receptance coupling method. International Journal of Advanced Manufacturing Technology 2008;35(11–12):1251–62. [9] Schmitz TL, Duncan GS. Three-component receptance coupling substructure analysis for tool point dynamics prediction. Journal of Manufacturing Science and Engineering 2005;127:781–9. [10] Burns TJ, Schmitz TL. A study of linear joint and tool models in spindleholder-tool receptance coupling. In: DETC2005: ASME international design engineering technical conferences and computers and information in engineering conference. 2005 [Code 66675]. [11] Kivanc EB, Budak E. Structural modelling of end mills for form error and stability analysis. Journal of Machine Tools & Manufacturing 2004;44: 1151–61. [12] Schmitz TL, Duncan GS. Receptance coupling for dynamics prediction of assemblies with coincident neutral axes. Journal of Sound and Vibration 2006;289:1045–65. [13] Movahhedy MR, Gerami JM. Prediction of spindle dynamics in milling by sub-structure coupling. International Journal of Machine Tools & Manufacture 2006;46:243–51. [14] Ahmadi K, Ahmadian H. Modelling machine tool dynamics using a distributed parameter tool–holder joint interface. International Journal of Machine Tools & Manufacture 2007;47(12–13):1916–28. [15] Erturk A, Ozguven HN, Budak E. Analytical modelling of spindle-tool dynamic on machine tools using Timoshenko beam model and receptance coupling for prediction of tool point FRF. International Journal of Machine Tools & Manufacture 2006;46:1901–12. [16] Erturk A, Ozguven HN, Budak E. Effect of bearing and interface dynamics on tool point FRF for chatter stability in machine tools by using a new analytical model for spindle-tool assemblies. International Journal of Machine Tools & Manufacture 2007;47:23–32. [17] Erturk A, Ozguven HN, Budak E. Selection of design and operational parameters in spindle-holder-tool assemblies for maximum chatter stability by using a
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Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Technical paper
Extending the inverse receptance coupling method for prediction of tool-holder joint dynamics in milling Mohammad Mahdi Rezaei ∗ , Mohammad R. Movahhedy, Hamed Moradi, Mohammad T. Ahmadian Center of Excellence in Design, Robotics and Automation (CEDRA), Department of Mechanical Engineering, Sharif University of Technology, Azadi Ave, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 4 April 2011 Received in revised form 3 October 2011 Accepted 21 November 2011 Available online 20 December 2011 Keywords: Inverse receptance coupling Joint model Tool-holder FRF Milling dynamics
a b s t r a c t Recently, receptance coupling substructure analysis (RCSA) is used for stability prediction of machine tools through its dynamic response determination. A major challenge is the proper modelling of the substructures joints and determination of their parameters. In this paper, a new approach for predicting tool tip FRF is presented. First, inverse RCSA formulation is extended so that the holder FRFs can be identified directly through experimental modal tests. The great advantage of this formulation is its implementation in arbitrary point numbers along joint length. Therefore, in comparison with previous inverse RCSA approaches, a more realistic joint model can be considered. In addition, due to applying the new approach, additional costly modal tests on the gauged tool are not required. This characteristic makes it possible to determine the holder FRFs without separating the tool; especially in situations where the holder end is inaccessible. The inclusion of joint parameters effect in the identified holder FRFs is another main advantage of such approach. Consequently, for identification of joint parameters, there is no need to use common error optimization based on fitting methods. The effect of overhang length is investigated through some analytical study and also experimental validation. Results show that the predicted tool tip FRF is exact in analytical case. Moreover, due to less noise effect, the predictions based on identified FRFs of longer tools are more accurate than the shorter ones (in experimental case). © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction Prediction and prevention of unstable chatter vibration is essential for achieving high quality and efficiency of machining processes. In the majority of methods used to find the stable free chatter regions of machining, finding the tool tip FRF is essential. End milling is one of the machining processes with most prone to the chatter. Budak and Altintas [1] presented a method to determine the stability lobe diagram in milling using tool tip FRF. In practice, experimental modal analysis is used to determine the tool tip FRF. However, using this method is time consuming and cumbersome, because the tool response changes as the tool is changed. In addition, for the cases where the experimental modal analysis is not possible such as flexible tools or micro milling, modelling is the only appropriate alternative. On the other hand, the modelling method based on only analytical results is a difficult task and often unreliable, especially for complex systems such as milling machine. An alternative is to couple the analytical response of the tool with the experimental response of the machine.
∗ Corresponding author. Tel.: +98 21 66165511; fax: +98 21 66000021. E-mail address: rezaei [email protected] (M.M. Rezaei).
The concept of receptance coupling has been extensively used for various applications. Wang and Liou [2] and Tsai and Chou [3] obtained the FRFs of a coupled system consisting of two planes from their separate FRFs. Ren and Beards [4] presented a more convenient approach of receptance coupling for application in linear joint modelling. In recent year, many works have been devoted to apply the RCSA in prediction of tool tip FRF in machining. Schmitz et al. [5] used RCSA to predict the FRFs of tool-spindle system. In their work, FRFs of the tool was found by analytical methods and was coupled with the experimental FRF of tool holder tip to find the assembly FRF. In this way it is possible to find the FRF of the assembled system when the tool is changed, without repetition of the experimental modal analysis. The main obstacle of this method is the joints modelling. Schmitz et al. [5] used a translational spring/damper to model the joint and identified its value by fitting the results of the experimental test and modelling. Liu and Ewins [6] and Park et al. [7] showed that neglecting the rotational FRFs through coupling process produces a large error in (especially in predicting FRFs of long length tools [7]). However, measuring rotational FRFs is difficult and expensive. To extract the rotational FRFs of tool holder, they proposed the inverse RCSA method. In this method, the direct FRF of holder set is determined by experimental modal test on gauged tool that has the same joint condition as the sample tool. Also, the tool rotational FRFs are identified through
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some experimental modal test on the tool connected to holder. Park and Chae [8] used this method for FRF prediction in modular tool applying receptance coupling method. Schmitz and Duncan [9] presented a three-component model including spindle, tool and tool holder and used the translational and rotational springs/dampers between the components. They found the values of the spring/damper by fitting the results of experimental test. Burns and Schmitz [10] investigated an analytical method to study the effect of tool overhang length on magnitude of joint parameters. They showed that the trend of joint parameters may be predictable only when a full connecting matrix is considered for it. Kivanc and Budak [11] considered complex geometry of fluted tool by a cylinder of equivalent diameter and used previous coupling method to predict tool-tip FRF. In the majority of mentioned works, the portion of tool located inside the tool holder is assumed as part of the holder while the joint is located between the overhang and tool holder substructure. If a multiple point model along the joint is applied, the prediction accuracy is increased. Schmitz and Duncan [12] developed the receptance coupling through multi point joint model. In other works, the accuracy of predictions was improved by modifying physics of the model through considering the joint along the contact interface. Movahhedy and Gerami [13] presented a model based on two spring/damper set at the joint ends. Consequently, it is not required to measure the rotational FRFs of the tool holder while the rotational modes are considered. In their work, a genetic algorithm (GA) was used to determine the joint stiffness and damper. A more complete model based on a continuous profile of the spring/damper along the tool holder interface was developed by Ahmadi and Ahmadian [14]. Erturk et al. [15–17] used a spring/damper set at the joint ends using the average values of the joint parameters provided in the literature. They demonstrated the capability of their model in optimizing the characteristics of the milling machine components. Namazi et al. [18] presented a joint model that included a set of translational and rotational springs/dampers along the joint. They found that the joint parameters extracted from a short length tool do not yield good results for prediction of the FRF of the longer overhang. Many of the RCSA works use fitting to experimental data as a way to extract joint parameters. However, this approach is not universally successful, because the search region for joint parameters is very wide and the process is time consuming. In addition, the results of the receptance coupling are very sensitive to joint parameters, and these values are not very reliable. Schmitz et al. [19] used a method based on contact model in commercial finite element software to determine joint parameters. In that method, simplifications and averaging was used to convert a complex 3D-model to a simple model of beam element. In their approach, successive averaging stages were performed which caused accuracy reduction. In this paper, a new method is proposed for direct determination of the tool tip FRF without identification of joint parameters by fitting method. In the first stage, the FRF of whole system (including tool and holder set) at tool tip is formulated as function of its substructures FRFs. Then, the inverse receptance coupling is extended for arbitrary point numbers in joint modelling. Using the extended procedure, the tool/holder set FRFs is obtained by subtraction of the analytical tool FRFs from whole assembly FRFs. The resulting FRFs represent the dynamics of both the tool holder and the joint. So, the whole assembly FRF at the tool tip with various overhang length is obtained through coupling of the extracted tool holder FRFs with analytical tool FRFs. In contrast to the previous works, using this method makes it possible to extract tool holder FRFs; especially when holder is inaccessible for direct measurements; e.g. at the joint surface inside the spindle bore. This paper is set as follows. In Section 2, the formulation of the extended RCSA and its inverse and their application in extracting
the tool holder FRFs is described. In Section 3, the accuracy and efficiency of the proposed approach is analytically investigated. In Section 4, experimental results are provided to evaluate the accuracy of the proposed method for various tool overhang lengths. Finally, conclusions are provided in Section 5. 2. The receptance coupling substructure analysis method and its inverse 2.1. Receptance coupling substructure analysis (RCSA) A simplified model of a milling machine is depicted in Fig. 1. In this model, the structure is constituted of two components: the tool (part B), and the tool holder and other parts of the structure (part A). The coupling process is performed through n points along tool holder/tool joint. Considering the FRFs of part A at n points, its n × n FRF matrix is written as:
⎧ ⎫ XA,1 ⎪ ⎪ ⎪ ⎨ XA,2 ⎪ ⎬
⎡
HA,11 ⎢ HA,21 =⎢ . .. ⎣ .. ⎪ ⎪ . ⎪ ⎪ ⎩ ⎭ XA,n HA,n1
HA,12 HA,22 .. . HA,n2
⎤⎧
··· ··· .. . ···
⎫
HA,1n ⎪ FA,1 ⎪ ⎪ ⎪ HA,2n ⎥ ⎨ FA,2 ⎬ ⎥ .. ⎦ .. ⎪ ⎪ . ⎪ ⎩ . ⎪ ⎭ FA,n HA,nn
(1)
where X and F denote the displacement and corresponding force vectors at each point, respectively. For instance, considering two degrees of freedom of lateral displacement and rotation degree for each point, the corresponding force vectors will be lateral force f and moment M. HA,ij is the FRF sub-matrix between points i and j of part A. For instance, the force vector and FRF sub matrix corresponding to displacements of point 1 due to applying a force at point 2 are:
XA,1 =
vA,1
,
A,1
HA,12 =
hA12,ff hA12,Mf
hA12,fM hA12,MM
,
FA,2 =
fA,2 MA,2
(2) Considering tool tip FRF at m point, the FRF matrix of part B and whole structure T will become:
⎧ ⎪ ⎨
XB,1 XB,2 . . ⎪ ⎩ . XB,n+m
⎫ ⎡ ⎪ ⎬ ⎢ =⎣ ⎪ ⎭
⎧ ⎫ XT,1 ⎪ ⎪ ⎪ ⎨ XT,2 ⎪ ⎬
⎡
HB,11 HB,21 . . .
HB,12 HB,22 . . .
HB,(n+m)1
HB,(n+m)2
HT,11
⎢ HT,21 =⎢ . .. ⎪ ⎪ ⎣ .. ⎪ ⎩ . ⎪ ⎭ XT,m
HT,m1
HT,12 HT,22 .. . HT,m2
··· ··· . . . ···
HB,1(n+n) HB,2(n+m) . . . HB,(n+m)(n+m)
⎤⎧ ⎪ ⎨ ⎥ ⎦ ⎪ ⎩
FB,1 FB,2 . . . FB,(n+m)
⎫ ⎪ ⎬ ⎪ ⎭
⎫ ⎤⎧ · · · HT,1m ⎪ FT,1 ⎪ ⎪ ⎪ · · · HT,2m ⎥ ⎨ FT,2 ⎬ ⎥ .. .. ⎦ .. ⎪ ⎪ . . ⎪ ⎩ . ⎪ ⎭ ···
HT,mm
(3)
(4)
FT,m
On the other hand, the FRF matrix of the whole structure can be obtained by coupling its substructure at n points along the joint through RCSA method. To simplify the formulation, points are divided to contact points with subscript C¯ and non-contact points ¯ so Eqs. (2), (3) and (4) are rewritten as: with subscript n; XA,¯c = SA,¯cc¯ FA,¯c
XB,¯c XB,n¯
=
SB,¯cc¯ SB,n¯ ¯c
SB,¯cn¯ SB,n¯ n¯
FB,¯c FB,n¯
XT,n¯ = ST,n¯ n¯ FT,n¯
(5) (6) (7)
At the points where RCSA is performed and at points on the tool, equilibrium and compatibility equations are described as follows:
FA,¯c + FB,¯c = 0 , XA,¯c = XB,¯c
FB,n¯ = FT,n¯ XB,n¯ = XT,n¯
(8)
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Fig. 1. Two part model of the milling machine including tool and tool holder.
Substituting the values of XA,¯c and XB,¯c from Eqs. (5) and (6) into Eq. (8) yields: −SA,¯cc¯ FB,¯c = SB,¯cc¯ FB,¯c + SB,¯cn¯ FT,n¯ ⇒ FB,¯c = −(SA,¯cc¯ + SB,¯cc¯ )−1 SB,¯cn¯ FT,n¯ (9) Substituting FB,¯c in Eq. (6), XT,n¯ at the end points of the tool is obtained as: −1 XT,n¯ = (SB,n¯ n¯ − SB,n¯ ¯ c (SA,¯c c¯ + SB,¯c c¯ ) SB,¯c n¯ )m×m FT,n¯
(10)
Therefore, according to Eq. (7) the obtained whole system FRFs in terms of its substructure FRFs is given as: −1 ST,n¯ n¯ = SB,n¯ n¯ − SB,n¯ ¯ c (SA,¯c c¯ + SB,¯c c¯ ) SB,¯c n¯
(11)
2.2. Extended inverse receptance coupling substructure analysis In this section, the inverse approach of RCSA is described by presenting the reverse of the above RCSA approach. If all substructure FRFs of a structure are known, the FRF matrix of the assembled structure can be determined through applying RCSA method. This requires that the joint parameters at the interface be known. To avoid determination of joint parameters by fitting methods, an alternative simple method is used. In this method, the tool holder FRFs are determined by subtracting of the analytically obtained tool FRFs from the FRFs of tool holder/tool assembly. The extracted FRFs of tool holder are different from those obtained by performing direct experimental modal test on tool holder, because the former includes the dynamics of the joint parameters between tool holder/tool. Then, this FRF can be directly used to predict assembly FRF for every combination of the same tool holder and any other tool, provided that the joint has the same condition. The details of the developed inverse RCSA method for extraction of the tool holder FRFs at n points along joint length by having the whole structure FRFs at m points on the tool (Fig. 1) is given below. Considering two degrees of freedom for each point of model, the FRF matrix of whole structure T is 2m × 2m. As shown in Fig. 1, n points along the joint are considered for coupling purpose; so the extracted FRF matrix A is a 2n × 2n matrix. Eq. (11) is used to determine the tool holder FRFs. In this equation the FRF matrices of SB,n¯ n¯ , SB,n¯ ¯ c , SB,¯c n¯ and SB,¯c c¯ are the tool FRFs that can be obtained analytically. The FRF matrix of ST,n¯ n¯ is known from experimental modal measurement at tool tip. The aim is to obtain the FRF matrix SA,¯cc¯
by solving Eq. (11). Although the solution of this equation seems to be nonlinear, it is made linear by changing the unknown SA,¯cc¯
to the new unknown matrix H = (SA,¯cc¯ + SB,¯cc¯ )−1 . For determining the FRF matrix at n points of joint interface, the number of measured FRFs of structure, 2m, must be at least equal to the desired number of FRFs, 2n. To simplify the calculation, it is assumed that the same number of measurements m as the number of coupling point n is made. If the measured FRFs of ST,n¯ n¯ are independent, there is a unique series for SA,¯cc¯ . Due to symmetry of the FRF matrices, the number of both independent equations and unknown variables is 2n × (2n + 1)/2. After determination of H, matrix SA,¯cc¯ is simply obtained as follows: H = (SA,¯cc¯ + SB,¯cc¯ )−1 ⇒ SA,¯cc¯ = H −1 − SB,¯cc¯
(12)
As mentioned before, the number of identified FRFs of SA,¯cc¯ is equal to the number of independent measured FRFs of ST,n¯ n¯ . The maximum number of independent measured FRFs is three. This is because, if the FRF sub-matrix at any point is known, the FRFs of tool at any other points can be obtained by applying RCSA method directly. In the considered model, each point has two degrees of freedom and consequently the sub-matrix at any point is 2 × 2; which due to symmetry has only three independent elements. In other words, maximum number of tool holder FRFs is three. Although, this sets a limit for extraction of tool holder FRFs, it also implies that having three FRFs of ST,n¯ n¯ is enough to extract tool holder FRFs so that exact FRFs at any point of structure can be predicted. In this paper, the coupling process is performed by using force-lateral displacement FRFs of two points along the joint length (Fig. 2). In some of the previous works [7,9], connection between two components was at a single point, and the FRFs at the connection point included rotational, translational and cross functions (Hff , Htt , Htf ). However, in following sections, the connection is established at two points (1 and 2) and the concerning FRFs are H11 , H12 , H22 (translational FRFs). It is shown that by considering two connection points, the effect of rotation is automatically included (as also discussed in [13]). The force-lateral displacement FRF matrix of components A, B and whole structure T, Fig. 2, are described as:
vA,1 vA,2
=
hA11,ff hA21,ff
hA12,ff hA22,ff
fA,1 fA,2
(13)
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Fig. 2. Two part model of milling machine used for FRF coupling at two points along the joint.
⎧ ⎫ ⎪ ⎨ vB,1 ⎪ ⎬
⎡
hB11,ff vB,2 ⎢ hB21,ff =⎣ hB31,ff ⎪ ⎪ ⎩ vB,3 ⎭ vB,4 hB41,ff
vT,1 vT,2
=
hT 11,ff hT 21,ff
hB12,ff hB22,ff hB32,ff hB42,ff hT 12,ff hT 22,ff
hB13,ff hB23,ff hB33,ff hB43,ff
fT,1 fT,2
hB14,ff hB24,ff hB34,ff hB44ff
⎫ ⎤⎧ ⎪ ⎨ fB,1 ⎪ ⎬ ⎥ fB,2 ⎦ f ⎪ ⎩ B,3 ⎪ ⎭
(14)
fB,4
(15)
where hAij,ff , hBij,ff and hTij,ff are the force/translational displacement FRFs between points i and j of parts A, B and T, respectively. According to Eq. (11), the FRF matrix of the whole structure T can be obtained through coupling of its substructures A and B. Substituting FRF matrix formulations from Eqs. (13), (14) and (15) into Eq. (11) yields:
hT 11,ff hT 21,ff
hT 12,ff hT 22,ff
=
hB34,ff hB44,ff
hB33,ff hB43,ff
×
˛ ˇ
ˇ
×
+
hB13,ff hB23,ff
hB31,ff hB41,ff hB14,ff hB24,ff
hB32,ff hB42,ff
(16)
where ˛, ˇ and are the elements of the unknown matrix H, given in Eq. (12). Making equal the matrix elements of both sides of Eq. (16), yields the set of equations as follows:
hB13,ff hB31,ff hB13,ff hB32,ff + hB23,ff hB31,ff hB23,ff hB32,ff hB13,ff hB14,ff hB13,ff hB24,ff + hB14,ff hB23,ff hB23,ff hB24,ff hB14,ff hB41,ff hB14,ff hB42,ff + hB24,ff hB41,ff hB24,ff hB42,ff
=
hB33,ff − hT 11,ff hB34,ff − hT 12,ff hB44,ff − hT 22,ff
˛ ˇ
(17)
Using Eq. (17) yields ˛, ˇ and , then the elements of the SA,¯cc¯ matrix are found through Eq. (12). 3. Numerical investigation In this section, numerical modelling is used to investigate the applicability and accuracy of the proposed inverse RCSA approach. The advantage of this approach in contrast to experimental analysis is the elimination of the sources of the experimental error especially associated with the measurement noise. In this investigation, a model of tool holder/tool assembly is considered as shown in
Fig. 3. A continuous distribution of spring/damper is considered to model the connection between tool holder and tool. Such a model has been used in the literature as one of the most precise models that can predict the behaviour of the tool holder/tool connection if the properties of spring and damper are determined properly. In this investigation, we have adopted these properties from [7]. The FRFs of the assembly is obtained through finite element method (FEM). Since the objective of this analysis is to evaluate the effectiveness of the RCSA and IRCSA methods, the Euler–Bernoulli beam elements are used for simplification. However it should be noted that for components with small slenderness ratio, more accurate models such as Timoshenko beam elements can be used. Then, the presented method is used for extraction of the FRFs of the tool holder. In the following, the effect of variation of tool overhang length on the extracted values of the tool holder FRFs (including joint parameters) is studied. Also, the extracted tool holder FRFs for a definite length of tool are used to predict the tool FRF with different overhang lengths. Fig. 4 shows the three tool assemblies with different overhang lengths while other parameters are considered equal (as given in Table 1). For each assembly, tool tip FRF at end points 1 and 2 is found analytically. Next, the tool holder FRFs (including joint parameters) is extracted using the proposed inverse RCSA method. Fig. 5 shows the extracted tool holder FRFs for the three assemblies. It is observed that the extracted FRFs coincide with each other. This confirms this fact that with similar joint conditions, tool overhang length has no effect on extracted holder FRFs. Next, we investigate that the any of the extracted tool holder FRFs can be used to predict the FRFs of other cases. In this investigation, cases A, B and C correspond to the extracted holder FRFs from short, medium and long tools, respectively. Fig. 6 shows the tool-tip FRF obtained by coupling the extracted tool holder FRFs of cases A, B and C for the tool overhang length of case A. Similarly, other extracted FRFs can be used to reconstruct the assembly FRFs for other cases with similar results. It is observed that the predicted FRF results coincide with each other and with the analytically obtained FRFs of the total assembly. Therefore, the predicted holder base FRFs for each tool is independent of the case from which it is extracted. This is because, in numerical analysis, linear modelling is considered for the tool/tool-holder connection. Therefore, increasing tool overhang length is in analogous with increasing in the dynamic loading which does not change the connection FRFs.
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203
Fig. 3. Continuous uniform distribution of spring/damper between tool and holder for analytical modelling of milling machine, tool holder and tool joint.
Fig. 4. Three tool samples with different overhang lengths. Table 1 The nominal specification of the model. Tool Et t Dt Lt1 Lt2 t
Tool holder 540 GPa 14400 kg/m3 20 mm 100 mm 50 mm 0.02
Eh h Lh1 Lh2 Dh1 Dh2 h
Contact parameter 200 GPa 7800 kg/m3 110 mm 70 mm 20 mm 40 mm 0.02
kc c
1.37 × 1012 N/m3 0.35
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M.M. Rezaei et al. / Journal of Manufacturing Processes 14 (2012) 199–207
Fig. 5. The extracted tool holder FRFs for three cases with different overhang lengths.
Fig. 6. The analytical and predicted tool tip FRF of short tool for three cases of extracted holder FRFs.
In addition, it is shown that the coupling of FRFs at just two points along the interface is sufficient for prediction of the tool tip FRF.
machine and experimental modal analysis is carried out. Also, other joint specifications such as the collet torque are not changed from one sample test to another. The FRFs of each tool at points of 3 and 4 (Fig. 7) are obtained through experimental modal analysis. According the proposed method in Section 2.2, for each of three tools, holder FRFs are extracted by subtracting tool analytical FRFs from the experimental FRFs of the assembled system. Extracted tool holder FRFs for each of three samples are shown in Figs. 8–10. It is observed that the extracted holder FRFs for long and medium length tools are close to each other while that for the short tool contains larger fluctuations. The source of these fluctuations may be attributed to the effect of experimental noise. It should be noted that in the IRCSA method, FRFs of one component are subtracted
4. Experimental investigation In this section, the accuracy and efficiency of the proposed method is investigated through comparing with experimental results. The experimental results are adopted from the previous work [7]. The effect of changing tool overhang length is studied while other parameters are constant. Three carbide cylinder blanks with specifications given in Fig. 7 and density of 14480 kg/m3 and modulus of elasticity of 580 GPa are considered. All samples with joint length of 20 mm are attached in a vertical three axes-milling
Fig. 7. Three blanks of different overhang lengths used for experimental investigation.
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Fig. 8. The direct FRF of point 1 (h11,ff ), extracted from three tools with different overhang lengths.
Fig. 9. The cross FRF of point 1 (h12,ff ), extracted from three tools with different overhang lengths.
Fig. 10. The direct FRF of point 2 (h22,ff ), extracted from three tools with different overhang lengths.
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Fig. 11. Tool tip FRF of long tool, predicted by extracted tool holder FRFs of different overhang lengths.
from those of total system. So, in the remained parts of signal (with small amplitudes), signal to noise ratio is decreasing. This phenomenon is more adverse around the resonance frequencies due to the existence of computational errors arisen by singularities (in inverse operations). Therefore, as the signal to noise ratio is increased, the final results are more reliable. Such behaviour, that the longer tools with larger signal amplitudes are less prone to noise, is observed in next results. Moreover, the relative distance between the points 3 and 4 (Fig. 7) can be important in some cases. Generally, by decreasing the distance between the measured points, the order of difference between measured signals approaches to that of noise signal. This matter is essentially observed in results of short tool. Under such conditions, to increase the accuracy of results, signal smoothing by using filter methods can be applied (e.g., [9]). Next, extracted holder FRFs for each of the cases short, medium and long cases, shown in Figs. 8, 9 and 10, are used to predict tool tip FRF for different overhang lengths. In these figures, case A, B and C correspond to the extracted holder FRFs of the short, medium and long tools, respectively. Fig. 11 shows the tool tip FRF of the long tool predicted by coupling its analytically obtained FRFs with extracted tool holder FRFs in each of the cases A, B and C. The predicted tool tip FRF have dominant frequencies at 400, 545, 770 and 970 Hz. Fig. 11 shows that the experimentally measured FRFs [7] matches with the predicted FRFs of the long length tool (case C) exactly. This confirms that using presented inverse RCSA method, leads to the desired tool holder FRFs as well. The predicted tool tip FRF of the long tool in the case B (related to extracted holder FRF of medium tool case) is close to the measured FRF. However, in peak frequencies, it predicts larger amplitudes than the measured ones. Again, less agreement is observed for case A, which was expected because its corresponding tool holder FRFs contained large fluctuations too. Fig. 12 shows the medium tool tip FRF for each cases of extracted tool holder FRFs. Similarly, the predicted tool tip FRF for case B coincides perfectly the measured FRF. The tool tip FRF of case C is close to the measured one. This confirms that the extracted holder FRFs for long tool can be used to predict the tool tip FRF of medium tool. For case A, predicted tool tip FRF has larger fluctuations again. Tool tip FRF for short length tool for each cases of extracted tool holder is shown in Fig. 13. It is observed that the predicted tool tip FRF in all cases are close to each other. Therefore it can be concluded that the extracted holder FRFs for the longer tool can be used in prediction of tool tip FRF of shorter tools. But prediction
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Fig. 12. Tool tip FRF of medium tool, predicted by extracted tool holder FRFs of different overhang lengths.
The great advantage of such approach is that joint parameters effect is reflected in the identified holder FRFs. Consequently, unlike the previous works, there is no need for identification of joint parameters through commonly tedious fitting methods. Another advantage of this modified formulation is the extraction of experimental holder set FRFs at arbitrary point numbers along joint length. It is shown that the coupling of FRFs at just two points along the interface is sufficient for prediction of the tool tip FRF. In addition, unlike the previous works using inverse RCSA method, additional modal tests on the gauged tool are not required. Specifically in situations that direct FRF measurement is impossible (when the holder end is inaccessible), using the proposed approach is more applicable. Some analytical investigations and experimental validations are performed to study the effect of overhang length on the accuracy of method predictions. In analytical case, exact predictions of tool tip FRF are obtained regardless of the overhang length of sample tool. According to experimental results, good predictions of assembly FRFs are achieved for sufficiently long tools. However, the results are less accurate for shorter tools, where the experimental noise effect is rather small. In future, this proposed method is developed to be capable of prediction for tool/holder cases with different joint parameters such as tool diameter and material, joint length and contact pressure.
References
Fig. 13. Tool tip FRF of short tool, predicted by extracted tool holder FRFs of different overhang lengths.
of long length tool tip FRF using extracted FRFs of shorter tools is less accurate. This inaccuracy depends partly on the order of the signal to noise ratio in measurements. For shorter tools with larger noise in measurements, less accurate results are obtained. Therefore, if the tools lengths are long enough, predictions are good. 5. Conclusions In this paper, prediction of tool tip FRF is performed through a new extended inverse RCSA method. This convenient approach overcomes the major challenge in the determination of joint parameters in substructures connections. Direct identification of holder FRFs through experimental modal tests are accomplished by extending an inverse RCSA formulation. In this method, the tool/holder set FRFs is obtained by subtraction of the analytical tool FRFs from whole assembly FRFs. Therefore, the tool tip FRF of other tools with different overhang lengths is predicted through coupling of the extracted tool holder FRFs with analytical tool FRFs.
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