- Email: [email protected]
An approach for measuring the FRF of machine tool structure without knowing any input force
International Journal of Machine Tools & Manufacture 86 (2014) 62–67
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Short Communication
An approach for measuring the FRF of machine tool structure without knowing any input force Xinyong Mao b, Bo Luo b, Bin Li a,n, Hui Cai b, Hongqi Liu b, Fangyu Pen b a
State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China National NC System Engineering Research Center, Huazhong University of Science and Technology, 1037 Luoyu Road, Hongshan District, Wuhan 430074, Hubei, China
b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 February 2014 Received in revised form 1 June 2014 Accepted 3 July 2014 Available online 14 July 2014
Measuring the dynamics of a machine tool is important for improving its processing or design. In general, the dynamics of the machine tool structure is identified by the experimental modal analysis approaches that require the measurement of both the input loadings and the corresponding structural responses. However, the primary limitation for this method is that the input loadings are difficult or impossible to be measured when the machine tool is under operational conditions. In this paper, a method that is based on random decrement technology was used to identify the operational modal parameters of a machine tool without the knowledge of any of the inputs. To estimate the frequency response functions, FRFs, a structural change method was proposed. The approach is based on the sensitivity of the eigenproperties to structural modifications caused by the drive positions. The proposed method was verified experimentally by traditional hammer tests. Because no elaborate excitation equipment is used, the dynamics of the machine tool structure with arbitrarily feed rate or working position can be easily identified using the proposed active excitation modal analysis method. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Frequency response functions Mode shape scaling Structural dynamics Machine tools
1. Introduction The quality and the geometrical and dimensional accuracy of machined parts depend greatly on the dynamics of the machine tool structure. Generally, there are two types of methods to estimate the dynamics of the entire machine tool structure. One method is based on the computer-aided engineering. Altintas et al. summarized the use of finite element analysis of the dynamic behavior of a machine tool structure in a review article [1]. However, it is difficult to calculate the dynamic parameters with high accuracy using this method because the dynamics of the joints are complex and abstruse [2]. Another method is experimental modal analysis (EMA), which is based on the performance of forced vibration tests in the laboratory, involving the measurements of the frequency response function and the impulse response function. However, the traditional EMA has certain limitations. For example: (1) FRF and IRF are highly difficult, or even impossible, to measure in the field, especially for large structures, and (2) in many industrial applications, the real operation conditions may differ significantly from those for lab testing with artificial excitation [3].
n Corresponding author. Tel.: þ 86 27 87542613 8428, mobile: þ 86 13507115177; fax: þ86 027 87540024. E-mail address: [email protected] (B. Li).
http://dx.doi.org/10.1016/j.ijmachtools.2014.07.004 0890-6955/& 2014 Elsevier Ltd. All rights reserved.
Operational modal analysis (OMA) is a powerful tool for estimating a structure's dynamics during operation conditions. OMA only uses the operational or natural responses of the structure without using any artificial excitation. Compared with EMA, the OMA method exhibits many advantages. For example: (1) the OMA is inexpensive and fast to conduct, it does not require elaborate excitation equipment and boundary condition simulation and (2) because the structures are characterized using real operation conditions, in the case of the existence of non-linear behavior, the obtained results are associated with realistic levels of vibration and not with artificially generated vibrations. Currently, several researchers have used OMA to estimate the dynamics of a machine tool and workpiece system by cutting force [4]. Nevertheless, there are three problems with these applications of the OMA method. (1) The estimated results were the dynamics of the cutting system, which may be different from the machine tool structure. (2) Because the input forces in the OMA are unknown, the mode shapes cannot be mass normalized, and only the un-scaled mode shapes can be determined for each mode. However, the scaling factors of the mode shapes must be known to assemble the FRF matrix of the machine tool from the modal parameters. (3) Identification of the dynamics in the entire machine work volume is difficult. In this paper, an active excitation modal analysis method (AEMA) is presented to estimate the dynamics of a machine tool structure. The proposed method can avoid the above mentioned problems.
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
63
where αr is the scaling factor. Then, the expression of the FRF matrix using unscaled mode shapes can be given by
2. Active excitation modal analysis 2.1. Theory of the active excitation modal analysis
n
½HðωÞ ¼ ∑
The active excitation modal analysis method uses the inertial force of the sliders during the acceleration or deceleration process to excite the machine tool structure [5]. Using the ball-screw driver system, the inertial force is applied to the machine tool. Because the movement is restricted in a small area, the variation of the dynamics due to the structural change is small enough to be ignored. To estimate the dynamic parameters, it is critical that the inertial force must excite all the natural frequencies of the structure in the bandwidth of interest. Some researchers have proved that the inertial forces are square-wave pulses when the slider accelerates or decelerates [6]. Therefore, in AEMA, a large accelerated speed and a small feed speed should be designed to obtain a high frequency range. The free decay response is another critical factor to estimate the modal parameters from the structure vibration. Once the free decay response function of a structure is acquired, the modal parameters of the structure, including natural frequencies, damping ratios, and mode shapes, can be identified using the time domain modal identification method, such as the eigensystem realization algorithm (ERA). For the active excitation modal analysis, the free decay response can be extracted by using the random decrement technology (RDT) [7].
r¼1
ω
2 r
α2r ηr ηΤr ω2 þi2ζ r ωr ω
ð3Þ
Thus, if the scaling factor for each mode is determined by a method complementary to OMA, the FRF matrix can be assembled although the inputs are unknown. In the last several years, several approaches have been proposed in the literature for scaling the modal shapes estimated by OMA. The methods based on dynamic modification that use the operational modal parameters of both the modified and unmodified structure. The dynamic modification is usually conducted by adding the masses or springs to the structure. However, these methods become inappropriate for a system such as a machine tool because it is inconvenient and difficult to add masses or springs to change the dynamics. This paper proposes a new method to estimate the scaling factor of the operational mode shape of the machine tool structure. The dynamic modification is made by the structural change of the machine tool. In this case, the entire machine tool structure can be divided into two major substructures: the immovable substructure that is composed of the bed and the column, the movable substructure consisting of the head stock, the slider, etc. Fig. 1 shows the simple discrete model of a structure composed of only the immovable component and the movable component at two different positions. The equations of motion of the structure at position #1 can be represented as X X X MA þ CA þ KA ¼0 x x x
2.2. Theory of the FRF assembling method When the modal space is used, the FRF matrix of the machine tool can be expressed by the sum of the contributions of the dynamic system modes [8]
ϕr ϕΤr 2 2 r ¼ 1 ωr ω þi2ζ r ωr ω n
½HðωÞ ¼ ∑
ð1Þ
Each mode is defined by a natural frequency ωr , a damping ratio ζ r and the mass normalized mode shape ϕr ηr In OMA, because the exciting forces are unknown, the mode shape cannot be normalized and only the unscaled mode shape can be determined for each mode. However, the relationship between the scaled mode shape ϕr and the unscaled mode shape ηr can be expressed by
ϕr ¼ αr ηr 2
K1 þ K2
6 K 6 2 6 6 ⋮ 6 6 0 6 6 6 ⋮ 6 6 0 6 KA ¼ 6 6 ⋮ 6 6 0 6 6 6 0 6 6 0 6 6 6 ⋮ 4 0
ð4Þ
where MA, KA and CA are the mass, damping and stiffness matrices, respectively, for the total DOFs of the structure at position #1. X is the displacement vector of the immovable component, and x is the displacement vector of the movable component. The mass and stiffness matrices can be further expressed as the following: 2 3 M1 6 7 M2 6 7 6 7 6 7 ⋱ 6 7 6 7 Mn 6 7 6 7 MA ¼ 6 ð5Þ 7 m 1 6 7 6 7 m2 6 7 6 7 6 7 ⋱ 4 5 mp and
ð2Þ K2
0
K2 þ K3 ⋱
0
0
0
0
0
0
0
0
K 3
0
0
0
0
0
0
0
0
⋱
⋱
⋮
⋮
⋮
⋮
⋮
⋮
⋮
0
Ka
K a þ K a þ 1 þ k1
Ka þ 1
0
0
0
k1
0
0
⋮ 0
⋮ 0
⋱ 0
⋱ Kb
⋱ Kb þ Kb þ 1
⋮ Kbþ 1
⋮
⋮ 0
⋮ 0
⋮ 0
⋮
⋮
⋮
⋱
⋱
⋱
⋱
⋮
⋮
⋮ 0
0
0
0
0
0
Kn
Kn
0
0
0
0
k1
0
0
0
0
k1 þk2
k2
0
0
0
0
0
0
0
0
k2
k2 þ k3
k3
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋱
⋱
0
0
0
0
0
0
0
0
0
kp
0
3
0 7 7 7 ⋮ 7 7 0 7 7 7 0 7 7 0 7 7 7 ⋮ 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 5 kp
ð6Þ
64
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
Fig. 1. Discrete models of the structure at different positions.
where M 1 ,M 2 ,…,M n and K 1 ,K 2 ,…,K n are the masses and stiffness coefficients, respectively, of the DOFs of the immovable component; m1 , m2 , ⋯, mp and k1 , k2 , …, kp are the masses and stiffness coefficients, respectively, of the DOFs of the movable component. The classical eigenvalue equation of Eq. (4) in the case of no damping or proportional damping is the following: ( ) ( ) MA
φr φr 2 ϕr ωr ¼ KA ϕr
ð7Þ Fig. 2. The active excitation modal analysis at three positions of the machine tool work volume.
and 9 8 φ > > > > > 1r > > = <φ > 2r ; φr ¼ ⋮ > > > > > > > > :φ ;
ϕr ¼
9 8 ϕ > > > > > 1r > > = < ϕ2r > ⋮ > > > > :ϕ
pr
nr
ð8Þ
> > > > ;
where φr is the scaled mode shape vector of the immovable component, and ϕr is the scaled mode shape vector of the movable component; ωr is the natural frequency of the structure at position #1. The equations of motion of the structure at position #2 are represented as the following:
MB
X0 x0
!
þ CB
X0 x0
!
X0
þ KB
x0
! ¼0
ð9Þ
where MB , CB and KB are the mass, damping and stiffness matrices, respectively, for the total DOFs of the structure at position #2. X0 is the displacement vector of the immovable component, and x0 is the displacement vector of the movable component. The classical eigenvalue equation of Eq. (9) can be expressed as follows: ( 0) ( 0) MB
φr φr 02 ϕ0r ωr ¼ KB ϕ0r
ð10Þ
and
φr ' ¼
8 0 9 φ1r > > > > > > > = <φ ' > 2r
> ⋮ > > > > > > ; : φ0 > nr
0
; ϕr ¼
8 0 9 ϕ > > > > > 01r > > = <ϕ > 2r
⋮ > > > > : ϕ0
pr
> > > > ;
ð11Þ
Fig. 3. The stabilization diagrams of the active excitation modal analysis at position #1. (a) The frequency identification. and (b) The damping identification. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
where φ0r is the scaled mode shape vector of the immovable 0 component, and ϕr is the scaled mode shape vector of the movable component. In practice, the dynamic stiffness of the head stock and the sliders are much larger than that of the frame (column and bed) of the machine tool [9]. Thus, for low-frequency structural modes, the movable component can be considered as a rigid body, and the following approximation can be applied:
ϕ1r ffi ϕ2r ffi ϕ3r ffi ⋯ ffi ϕpr ffi φar ϕ
0 1r ffi
ϕ
0 2r ffi
0 3r ffi ⋯ ffi
ϕ
ϕ
0 pr ffi
ð12Þ
φ
0 br
ð13Þ
Table 1 Modal parameters estimated by active excitation modal analysis at different positions. Position #1
Position #2
Position #3
Position #4
Mode #1
ω1 (Hz) ζ1 (%)
16.73 3.43
15.59 3.36
16.46 3.40
15.56 3.39
Mode #2
ω2 (Hz) ζ2 (%)
49.83 5.88
49.69 5.66
46.42 5.08
46.51 5.37
Mode #3
ω3 (Hz) ζ3 (%)
66.79 5.49
67.03 5.13
62.31 5.32
62.36 5.35
65
Assuming that the structural change on the mode shapes of the immovable component is negligible, then the scaled mode shape vectors of the immovable component at different positions are similar
φr ffi φ0r
ð14Þ
Subtracting Eqs. (7) and (10), we obtain the following: ( ) ( 0) ( ) ( 0) MA
φr φr φr φr 2 02 ϕr ωr MB ϕ0r ωr ¼ KA ϕr KB ϕ0r
ð15Þ
Substituting Eqs. (12) and (13) into the right hand of Eq. (15) yields the following: ( ) ( 0) MA
φr φr 2 02 ϕr ωr ¼ MB ϕ0r ωr
ð16Þ
n o T Premultiplying Eq. (16) by φr T ; ϕr , we obtain the following: ( φ ) ( φ0 ) n o M n o M r r Τ 2 T 02 φΤr ; ϕTr ω ¼ φ ; ϕ r r r ϕr ϕ0r ωr ð17Þ m m where M represents the mass matrix of the immovable component, and m represents the mass matrix of the movable component. Eq. (17) can be easily converted as the following: Τ 0 02 φΤr Mφr ω2r þ ϕΤr mϕr ω2r ¼ φΤr Mφ0r ω02 r þ ϕr mϕr ωr
Fig. 4. The power spectral density functions of the column at different positions.
ð18Þ
66
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
The unscaled mode shape μr and the scaled mode shape ϕr have the following relations:
ϕr ¼ αr μr
ð19Þ
ϕ0r ¼ αr μ0r
ð20Þ
Substituting Eqs. (19) and (20) into Eq. (18) yields the following: Τ 02 Τ 2 α2r μΤr mμr ω2r α2r μΤr mμ0r ω02 r ¼ φr Mφr ωr φr Mφr ωr
ð21Þ
According to Coppotelli [8], φΤr Μφr ¼ 1, then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ω02 r ωr αr ¼ Τ 0 ω2r μΤr mμr ω02 r μr M μr
ð22Þ
where μr and μ0r represent the mode shapes of the movable substructure at positions #1 and #2, respectively. Therefore, if we modify the dynamic behavior of the machine tool structure by changing its tool positions and perform operational modal tests at different positions, the scaling factors can be estimated from the operational modal parameters of both the modified and unmodified structure by Eq. (22). 3. Experimental validation 3.1. Experiment setup In the experiment, the worktable was restricted in a small fixed area, accelerated and decelerated randomly in the X-direction. To excite the machine tool structure, the feed speed of the worktable is 1000 mm/min, and the accelerated velocity is 1040 mm/s2. The mean value of the time duration between two adjacent accelerations is 0.05 s [5]. To achieve FRF, four positions, as shown in Fig. 2, were chosen to perform the active excitation modal analysis. Position #1 is the original position; position #2 is when the headstock has moved in the Z-direction by an amount of 0.5 m; position #3 is when the worktable moved in the X-direction by an amount of 0.5 m; and in position #4, the headstock moved in the Z-direction by an amount of 0.5 m and the worktable moved in the X-direction by an amount of 0.5 m. The mass of the headstock and worktable are 595 kg and 503 kg, respectively. As baseline experiments, traditional hammer test were conducted at two positions: one was the original position of the AEMA, and the other was position #4. 3.2. Active excitation modal analysis of the machine tool structure Fig. 3a and b shows the frequency and damping ratio stability diagrams of the active excitation modal analysis in the X-direction at position #1, respectively. The curve in Fig. 3a is the power spectral density function of the column in the X-direction. In these two figures, the model orders of the modal analysis algorithm are displayed on the ordinate axis and the calculated results are displayed on the abscissa axis [5]. There are six symbol clusters in Fig. 3a and three clusters in Fig. 3b. The calculated damping ratios of the modes in the bandwidth of 20–40 Hz are represented by the blue symbols in Fig. 3b. Note that these symbols are not convergent, which means the modes are spurious modes calculated from the noise. Therefore, there are three modes of the machine tool in the X-direction in the bandwidth of 0–75 Hz, and the natural frequencies and damping ratios are listed in Table 1. The power spectral density of the machine tool at positions #1, #2, #3 and #4 are compared in Fig. 4. It can be easily observed that only the first natural frequency is changed by the position of the headstock. For the second and third torsional modes, the natural frequencies were only influenced by the position of the worktable.
Fig. 5. Comparisons between the experimental FRFs and the assembled FRFs at positions #1 and #2.
The modal parameters estimated by active excitation modal analysis at the four positions are listed in Table 1. According to Eq. (22), the scaling factor for each mode can be calculated by using the rigid mode shapes of the headstock and the worktable. The calculated scaling factors for modes #1, #2, and #3 are 121.2, 1075, and 2162.5, respectively. Next, the FRF matrix of the machine tool can be assembled by substituting the scaling factors and the operational modal parameters of the machine tool into Eq. (3). Fig. 5 presents the comparisons of the experimental FRF and the assembled FRF of the machine tool at positions #1 and #4. As evident in the figures, the amplitudes of the assembled FRF are well matched with those of the experimental FRF, which means that the scaling factors are precisely calculated. However, for modes #1 and #3, the natural frequencies estimated by active excitation modal analysis are smaller than those of the hammer tests by approximately 1 Hz and 5 Hz, respectively. These results illustrate that the operational conditions may influence the dynamics of the machine tool structure.
4. Conclusions Traditional experimental modal analysis (EMA) of a machine tool structure is usually performed by artificial excitation of a structure and measuring input forces and output responses simultaneously. However, conventional EMA has several limitations, such as the test structure must be in a static condition and the requirement for a controllable input excitation source. With respect to the EMA techniques, the operational modal analysis (OMA) makes use of natural or operating loads to excite a structure, resulting in an easier method for characterizing the
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
dynamic behavior of the structure in real operative conditions. Nevertheless, because the input forces in OMA are unknown, only the unscaled mode shapes can be determined for each mode; as a result, the frequency response function is unknown. This paper presented an active excitation modal analysis (AEMA) method for estimating the modal parameters of a machine tool structure. Through this method, the machine tool is excited by the inertial force of random movements of the drive system, and the modal parameters can be identified by measuring only the responses of the structure. Using the dynamic modification by changing the drive positions, the mode shapes are scaled, and then the frequency response functions of the machine tool are assembled. The proposed method was experimentally validated, and the validation results indicated high accuracy and reliability. Because no artificial excitation is needed, this method can easily identify the spatial dynamics of a machine tool structure in the entire machine work volume. Moreover, the dynamics of the feed drive system under different feed rates can also be identified by the proposed method. Acknowledgments The research is supported by the National Natural Science Foundation of China under Grant no. 51375193 and the Science
67
and Technology Major Special Project of China under Grant no. 2011CB706803.
References [1] Y. Altintas, C. Brecher, M. Weck, S. Witt, Virtual machine tool, CIRP Ann. – Manuf. Technol. 54 (2005) 115–138. [2] Y. Altintas, A. Verl, C. Brecher, L. Uriarte, G. Pritschow, Machine tool feed drives, CIRP Ann. – Manuf. Technol. 60 (2011) 779–796. [3] S. Smith, J. Snyder, A cutting performance based template for spindle dynamics, CIRP Ann. – Manuf. Technol. 50 (2001) 259–262. [4] I. Zaghbani, V. Songmene, Estimation of machine-tool dynamic parameters during machining operation through operational modal analysis, Int. J. Mach. Tools Manuf. 49 (2009) 947–957. [5] B. Li, B. Luo, X. Mao, H. Cai, F. Peng, H. Liu, A new approach to identifying the dynamic behavior of CNC machine tools with respect to different worktable feed speeds, Int. J. Mach. Tools Manuf. 72 (2013) 73–84. [6] G. Feng, Y. Pan, Investigation of ball screw preload variation based on dynamic modeling of a preload adjustable feed-drive system and spectrum analysis of ball-nuts sensed vibration signals, Int. J. Mach. Tools Manuf. 52 (2012) 85–96. [7] S.R. Ibrahim, Random decrement technique for modal identification of structures, J. Spacecr. Rockets 14 (1977) 696–700. [8] G. Coppotelli, On the estimate of the FRFs from operational data, Mech. Syst. Signal Process. 23 (2009) 288–299. [9] C. Yuan Lin, J. Pin Hung, T. Liang Lo, Effect of preload of linear guides on dynamic characteristics of a vertical column-spindle system, Int. J. Mach. Tools Manuf. 50 (2010) 741–746.
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Short Communication
An approach for measuring the FRF of machine tool structure without knowing any input force Xinyong Mao b, Bo Luo b, Bin Li a,n, Hui Cai b, Hongqi Liu b, Fangyu Pen b a
State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China National NC System Engineering Research Center, Huazhong University of Science and Technology, 1037 Luoyu Road, Hongshan District, Wuhan 430074, Hubei, China
b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 February 2014 Received in revised form 1 June 2014 Accepted 3 July 2014 Available online 14 July 2014
Measuring the dynamics of a machine tool is important for improving its processing or design. In general, the dynamics of the machine tool structure is identified by the experimental modal analysis approaches that require the measurement of both the input loadings and the corresponding structural responses. However, the primary limitation for this method is that the input loadings are difficult or impossible to be measured when the machine tool is under operational conditions. In this paper, a method that is based on random decrement technology was used to identify the operational modal parameters of a machine tool without the knowledge of any of the inputs. To estimate the frequency response functions, FRFs, a structural change method was proposed. The approach is based on the sensitivity of the eigenproperties to structural modifications caused by the drive positions. The proposed method was verified experimentally by traditional hammer tests. Because no elaborate excitation equipment is used, the dynamics of the machine tool structure with arbitrarily feed rate or working position can be easily identified using the proposed active excitation modal analysis method. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Frequency response functions Mode shape scaling Structural dynamics Machine tools
1. Introduction The quality and the geometrical and dimensional accuracy of machined parts depend greatly on the dynamics of the machine tool structure. Generally, there are two types of methods to estimate the dynamics of the entire machine tool structure. One method is based on the computer-aided engineering. Altintas et al. summarized the use of finite element analysis of the dynamic behavior of a machine tool structure in a review article [1]. However, it is difficult to calculate the dynamic parameters with high accuracy using this method because the dynamics of the joints are complex and abstruse [2]. Another method is experimental modal analysis (EMA), which is based on the performance of forced vibration tests in the laboratory, involving the measurements of the frequency response function and the impulse response function. However, the traditional EMA has certain limitations. For example: (1) FRF and IRF are highly difficult, or even impossible, to measure in the field, especially for large structures, and (2) in many industrial applications, the real operation conditions may differ significantly from those for lab testing with artificial excitation [3].
n Corresponding author. Tel.: þ 86 27 87542613 8428, mobile: þ 86 13507115177; fax: þ86 027 87540024. E-mail address: [email protected] (B. Li).
http://dx.doi.org/10.1016/j.ijmachtools.2014.07.004 0890-6955/& 2014 Elsevier Ltd. All rights reserved.
Operational modal analysis (OMA) is a powerful tool for estimating a structure's dynamics during operation conditions. OMA only uses the operational or natural responses of the structure without using any artificial excitation. Compared with EMA, the OMA method exhibits many advantages. For example: (1) the OMA is inexpensive and fast to conduct, it does not require elaborate excitation equipment and boundary condition simulation and (2) because the structures are characterized using real operation conditions, in the case of the existence of non-linear behavior, the obtained results are associated with realistic levels of vibration and not with artificially generated vibrations. Currently, several researchers have used OMA to estimate the dynamics of a machine tool and workpiece system by cutting force [4]. Nevertheless, there are three problems with these applications of the OMA method. (1) The estimated results were the dynamics of the cutting system, which may be different from the machine tool structure. (2) Because the input forces in the OMA are unknown, the mode shapes cannot be mass normalized, and only the un-scaled mode shapes can be determined for each mode. However, the scaling factors of the mode shapes must be known to assemble the FRF matrix of the machine tool from the modal parameters. (3) Identification of the dynamics in the entire machine work volume is difficult. In this paper, an active excitation modal analysis method (AEMA) is presented to estimate the dynamics of a machine tool structure. The proposed method can avoid the above mentioned problems.
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
63
where αr is the scaling factor. Then, the expression of the FRF matrix using unscaled mode shapes can be given by
2. Active excitation modal analysis 2.1. Theory of the active excitation modal analysis
n
½HðωÞ ¼ ∑
The active excitation modal analysis method uses the inertial force of the sliders during the acceleration or deceleration process to excite the machine tool structure [5]. Using the ball-screw driver system, the inertial force is applied to the machine tool. Because the movement is restricted in a small area, the variation of the dynamics due to the structural change is small enough to be ignored. To estimate the dynamic parameters, it is critical that the inertial force must excite all the natural frequencies of the structure in the bandwidth of interest. Some researchers have proved that the inertial forces are square-wave pulses when the slider accelerates or decelerates [6]. Therefore, in AEMA, a large accelerated speed and a small feed speed should be designed to obtain a high frequency range. The free decay response is another critical factor to estimate the modal parameters from the structure vibration. Once the free decay response function of a structure is acquired, the modal parameters of the structure, including natural frequencies, damping ratios, and mode shapes, can be identified using the time domain modal identification method, such as the eigensystem realization algorithm (ERA). For the active excitation modal analysis, the free decay response can be extracted by using the random decrement technology (RDT) [7].
r¼1
ω
2 r
α2r ηr ηΤr ω2 þi2ζ r ωr ω
ð3Þ
Thus, if the scaling factor for each mode is determined by a method complementary to OMA, the FRF matrix can be assembled although the inputs are unknown. In the last several years, several approaches have been proposed in the literature for scaling the modal shapes estimated by OMA. The methods based on dynamic modification that use the operational modal parameters of both the modified and unmodified structure. The dynamic modification is usually conducted by adding the masses or springs to the structure. However, these methods become inappropriate for a system such as a machine tool because it is inconvenient and difficult to add masses or springs to change the dynamics. This paper proposes a new method to estimate the scaling factor of the operational mode shape of the machine tool structure. The dynamic modification is made by the structural change of the machine tool. In this case, the entire machine tool structure can be divided into two major substructures: the immovable substructure that is composed of the bed and the column, the movable substructure consisting of the head stock, the slider, etc. Fig. 1 shows the simple discrete model of a structure composed of only the immovable component and the movable component at two different positions. The equations of motion of the structure at position #1 can be represented as X X X MA þ CA þ KA ¼0 x x x
2.2. Theory of the FRF assembling method When the modal space is used, the FRF matrix of the machine tool can be expressed by the sum of the contributions of the dynamic system modes [8]
ϕr ϕΤr 2 2 r ¼ 1 ωr ω þi2ζ r ωr ω n
½HðωÞ ¼ ∑
ð1Þ
Each mode is defined by a natural frequency ωr , a damping ratio ζ r and the mass normalized mode shape ϕr ηr In OMA, because the exciting forces are unknown, the mode shape cannot be normalized and only the unscaled mode shape can be determined for each mode. However, the relationship between the scaled mode shape ϕr and the unscaled mode shape ηr can be expressed by
ϕr ¼ αr ηr 2
K1 þ K2
6 K 6 2 6 6 ⋮ 6 6 0 6 6 6 ⋮ 6 6 0 6 KA ¼ 6 6 ⋮ 6 6 0 6 6 6 0 6 6 0 6 6 6 ⋮ 4 0
ð4Þ
where MA, KA and CA are the mass, damping and stiffness matrices, respectively, for the total DOFs of the structure at position #1. X is the displacement vector of the immovable component, and x is the displacement vector of the movable component. The mass and stiffness matrices can be further expressed as the following: 2 3 M1 6 7 M2 6 7 6 7 6 7 ⋱ 6 7 6 7 Mn 6 7 6 7 MA ¼ 6 ð5Þ 7 m 1 6 7 6 7 m2 6 7 6 7 6 7 ⋱ 4 5 mp and
ð2Þ K2
0
K2 þ K3 ⋱
0
0
0
0
0
0
0
0
K 3
0
0
0
0
0
0
0
0
⋱
⋱
⋮
⋮
⋮
⋮
⋮
⋮
⋮
0
Ka
K a þ K a þ 1 þ k1
Ka þ 1
0
0
0
k1
0
0
⋮ 0
⋮ 0
⋱ 0
⋱ Kb
⋱ Kb þ Kb þ 1
⋮ Kbþ 1
⋮
⋮ 0
⋮ 0
⋮ 0
⋮
⋮
⋮
⋱
⋱
⋱
⋱
⋮
⋮
⋮ 0
0
0
0
0
0
Kn
Kn
0
0
0
0
k1
0
0
0
0
k1 þk2
k2
0
0
0
0
0
0
0
0
k2
k2 þ k3
k3
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋱
⋱
0
0
0
0
0
0
0
0
0
kp
0
3
0 7 7 7 ⋮ 7 7 0 7 7 7 0 7 7 0 7 7 7 ⋮ 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 5 kp
ð6Þ
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X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
Fig. 1. Discrete models of the structure at different positions.
where M 1 ,M 2 ,…,M n and K 1 ,K 2 ,…,K n are the masses and stiffness coefficients, respectively, of the DOFs of the immovable component; m1 , m2 , ⋯, mp and k1 , k2 , …, kp are the masses and stiffness coefficients, respectively, of the DOFs of the movable component. The classical eigenvalue equation of Eq. (4) in the case of no damping or proportional damping is the following: ( ) ( ) MA
φr φr 2 ϕr ωr ¼ KA ϕr
ð7Þ Fig. 2. The active excitation modal analysis at three positions of the machine tool work volume.
and 9 8 φ > > > > > 1r > > = <φ > 2r ; φr ¼ ⋮ > > > > > > > > :φ ;
ϕr ¼
9 8 ϕ > > > > > 1r > > = < ϕ2r > ⋮ > > > > :ϕ
pr
nr
ð8Þ
> > > > ;
where φr is the scaled mode shape vector of the immovable component, and ϕr is the scaled mode shape vector of the movable component; ωr is the natural frequency of the structure at position #1. The equations of motion of the structure at position #2 are represented as the following:
MB
X0 x0
!
þ CB
X0 x0
!
X0
þ KB
x0
! ¼0
ð9Þ
where MB , CB and KB are the mass, damping and stiffness matrices, respectively, for the total DOFs of the structure at position #2. X0 is the displacement vector of the immovable component, and x0 is the displacement vector of the movable component. The classical eigenvalue equation of Eq. (9) can be expressed as follows: ( 0) ( 0) MB
φr φr 02 ϕ0r ωr ¼ KB ϕ0r
ð10Þ
and
φr ' ¼
8 0 9 φ1r > > > > > > > = <φ ' > 2r
> ⋮ > > > > > > ; : φ0 > nr
0
; ϕr ¼
8 0 9 ϕ > > > > > 01r > > = <ϕ > 2r
⋮ > > > > : ϕ0
pr
> > > > ;
ð11Þ
Fig. 3. The stabilization diagrams of the active excitation modal analysis at position #1. (a) The frequency identification. and (b) The damping identification. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
where φ0r is the scaled mode shape vector of the immovable 0 component, and ϕr is the scaled mode shape vector of the movable component. In practice, the dynamic stiffness of the head stock and the sliders are much larger than that of the frame (column and bed) of the machine tool [9]. Thus, for low-frequency structural modes, the movable component can be considered as a rigid body, and the following approximation can be applied:
ϕ1r ffi ϕ2r ffi ϕ3r ffi ⋯ ffi ϕpr ffi φar ϕ
0 1r ffi
ϕ
0 2r ffi
0 3r ffi ⋯ ffi
ϕ
ϕ
0 pr ffi
ð12Þ
φ
0 br
ð13Þ
Table 1 Modal parameters estimated by active excitation modal analysis at different positions. Position #1
Position #2
Position #3
Position #4
Mode #1
ω1 (Hz) ζ1 (%)
16.73 3.43
15.59 3.36
16.46 3.40
15.56 3.39
Mode #2
ω2 (Hz) ζ2 (%)
49.83 5.88
49.69 5.66
46.42 5.08
46.51 5.37
Mode #3
ω3 (Hz) ζ3 (%)
66.79 5.49
67.03 5.13
62.31 5.32
62.36 5.35
65
Assuming that the structural change on the mode shapes of the immovable component is negligible, then the scaled mode shape vectors of the immovable component at different positions are similar
φr ffi φ0r
ð14Þ
Subtracting Eqs. (7) and (10), we obtain the following: ( ) ( 0) ( ) ( 0) MA
φr φr φr φr 2 02 ϕr ωr MB ϕ0r ωr ¼ KA ϕr KB ϕ0r
ð15Þ
Substituting Eqs. (12) and (13) into the right hand of Eq. (15) yields the following: ( ) ( 0) MA
φr φr 2 02 ϕr ωr ¼ MB ϕ0r ωr
ð16Þ
n o T Premultiplying Eq. (16) by φr T ; ϕr , we obtain the following: ( φ ) ( φ0 ) n o M n o M r r Τ 2 T 02 φΤr ; ϕTr ω ¼ φ ; ϕ r r r ϕr ϕ0r ωr ð17Þ m m where M represents the mass matrix of the immovable component, and m represents the mass matrix of the movable component. Eq. (17) can be easily converted as the following: Τ 0 02 φΤr Mφr ω2r þ ϕΤr mϕr ω2r ¼ φΤr Mφ0r ω02 r þ ϕr mϕr ωr
Fig. 4. The power spectral density functions of the column at different positions.
ð18Þ
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X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
The unscaled mode shape μr and the scaled mode shape ϕr have the following relations:
ϕr ¼ αr μr
ð19Þ
ϕ0r ¼ αr μ0r
ð20Þ
Substituting Eqs. (19) and (20) into Eq. (18) yields the following: Τ 02 Τ 2 α2r μΤr mμr ω2r α2r μΤr mμ0r ω02 r ¼ φr Mφr ωr φr Mφr ωr
ð21Þ
According to Coppotelli [8], φΤr Μφr ¼ 1, then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ω02 r ωr αr ¼ Τ 0 ω2r μΤr mμr ω02 r μr M μr
ð22Þ
where μr and μ0r represent the mode shapes of the movable substructure at positions #1 and #2, respectively. Therefore, if we modify the dynamic behavior of the machine tool structure by changing its tool positions and perform operational modal tests at different positions, the scaling factors can be estimated from the operational modal parameters of both the modified and unmodified structure by Eq. (22). 3. Experimental validation 3.1. Experiment setup In the experiment, the worktable was restricted in a small fixed area, accelerated and decelerated randomly in the X-direction. To excite the machine tool structure, the feed speed of the worktable is 1000 mm/min, and the accelerated velocity is 1040 mm/s2. The mean value of the time duration between two adjacent accelerations is 0.05 s [5]. To achieve FRF, four positions, as shown in Fig. 2, were chosen to perform the active excitation modal analysis. Position #1 is the original position; position #2 is when the headstock has moved in the Z-direction by an amount of 0.5 m; position #3 is when the worktable moved in the X-direction by an amount of 0.5 m; and in position #4, the headstock moved in the Z-direction by an amount of 0.5 m and the worktable moved in the X-direction by an amount of 0.5 m. The mass of the headstock and worktable are 595 kg and 503 kg, respectively. As baseline experiments, traditional hammer test were conducted at two positions: one was the original position of the AEMA, and the other was position #4. 3.2. Active excitation modal analysis of the machine tool structure Fig. 3a and b shows the frequency and damping ratio stability diagrams of the active excitation modal analysis in the X-direction at position #1, respectively. The curve in Fig. 3a is the power spectral density function of the column in the X-direction. In these two figures, the model orders of the modal analysis algorithm are displayed on the ordinate axis and the calculated results are displayed on the abscissa axis [5]. There are six symbol clusters in Fig. 3a and three clusters in Fig. 3b. The calculated damping ratios of the modes in the bandwidth of 20–40 Hz are represented by the blue symbols in Fig. 3b. Note that these symbols are not convergent, which means the modes are spurious modes calculated from the noise. Therefore, there are three modes of the machine tool in the X-direction in the bandwidth of 0–75 Hz, and the natural frequencies and damping ratios are listed in Table 1. The power spectral density of the machine tool at positions #1, #2, #3 and #4 are compared in Fig. 4. It can be easily observed that only the first natural frequency is changed by the position of the headstock. For the second and third torsional modes, the natural frequencies were only influenced by the position of the worktable.
Fig. 5. Comparisons between the experimental FRFs and the assembled FRFs at positions #1 and #2.
The modal parameters estimated by active excitation modal analysis at the four positions are listed in Table 1. According to Eq. (22), the scaling factor for each mode can be calculated by using the rigid mode shapes of the headstock and the worktable. The calculated scaling factors for modes #1, #2, and #3 are 121.2, 1075, and 2162.5, respectively. Next, the FRF matrix of the machine tool can be assembled by substituting the scaling factors and the operational modal parameters of the machine tool into Eq. (3). Fig. 5 presents the comparisons of the experimental FRF and the assembled FRF of the machine tool at positions #1 and #4. As evident in the figures, the amplitudes of the assembled FRF are well matched with those of the experimental FRF, which means that the scaling factors are precisely calculated. However, for modes #1 and #3, the natural frequencies estimated by active excitation modal analysis are smaller than those of the hammer tests by approximately 1 Hz and 5 Hz, respectively. These results illustrate that the operational conditions may influence the dynamics of the machine tool structure.
4. Conclusions Traditional experimental modal analysis (EMA) of a machine tool structure is usually performed by artificial excitation of a structure and measuring input forces and output responses simultaneously. However, conventional EMA has several limitations, such as the test structure must be in a static condition and the requirement for a controllable input excitation source. With respect to the EMA techniques, the operational modal analysis (OMA) makes use of natural or operating loads to excite a structure, resulting in an easier method for characterizing the
X. Mao et al. / International Journal of Machine Tools & Manufacture 86 (2014) 62–67
dynamic behavior of the structure in real operative conditions. Nevertheless, because the input forces in OMA are unknown, only the unscaled mode shapes can be determined for each mode; as a result, the frequency response function is unknown. This paper presented an active excitation modal analysis (AEMA) method for estimating the modal parameters of a machine tool structure. Through this method, the machine tool is excited by the inertial force of random movements of the drive system, and the modal parameters can be identified by measuring only the responses of the structure. Using the dynamic modification by changing the drive positions, the mode shapes are scaled, and then the frequency response functions of the machine tool are assembled. The proposed method was experimentally validated, and the validation results indicated high accuracy and reliability. Because no artificial excitation is needed, this method can easily identify the spatial dynamics of a machine tool structure in the entire machine work volume. Moreover, the dynamics of the feed drive system under different feed rates can also be identified by the proposed method. Acknowledgments The research is supported by the National Natural Science Foundation of China under Grant no. 51375193 and the Science
67
and Technology Major Special Project of China under Grant no. 2011CB706803.
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