Transactions on Engineering Sciences vol 44, © 2003 WIT Press, www.witpress.com, ISSN 1743-3533

Evaluating the structural dynamics of a vertical milling machine M.H.N. Widiyarto, D.G. Ford & C. Pislaru Ultra Precision Engineering Centre, School of Computing and Engineering, University of Huddersfield, United Kingdom.

Abstract Many CNC machine tools are used in general-purpose machining environments, resulting in a wide variety of machining conditions being present at the cutting tool. The requirement for guide-way to bearing clearances and limited mass of the structural elements results in a limited stiffness between the cutter and workpiece. This, coupled with the inertial effect of the structural element masses and drive-system, can lead to complex vibration modes that are affected by the axis positions and machine tool configuration. Therefore, the study of the dynamic behaviour of the machme is important in determining the response of the machine's structure to an external force. In this paper, the results and the setup of an experimental modal analysis performed on a vertical milling m a c h e at the Ultra Precision Engineering Centre of the University of Huddersfield are presented. T h s work will be used as the foundation to carry out further activities into the modelling and correction of vibration-induced machining errors on CNC machine tools.

1 Introduction Modal analysis is the study of the dynamic behaviour of a structure which aims to model the structure in terms of its modes of vibration or the modal properties, i.e. the natural frequencies, damping and mode shapes. The mode of vibration of a structure is a global property which can be measured from practically any point of the structure is a manifestation of energy trapped w i t h the boundaries of the structure that cannot be readily dissipated. Modal analysis is particularly suited to large, complex and non-homogeneous structures as opposed to other types of

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analysis such as lumped parameter and wave equation modeling, which are more suitable for simple rigid and elastic bodies respectively. Modal analysis of a structure can be performed by two methods: theoretical and experimental. Theoretical modal analysis is most commonly carried out using the finite element modeling techmque, whereas the experimental method commonly used is the modal testing techmque [2]. Both techniques complement each other in terms of model validation and structural dynamic prediction. Experimental modal analysis is often used for the validation and modification of a finite element model and, in turn, the validated model is used for the prediction of out-of-band modes or the structural dynamics of the structure. Experimental modal analysis is generally divided into two parts: modal testing and modal parameter extraction. Modal testing requires the measurement of the response of the structure for a measured input force. T h s type of measurement is also known as transfer function or frequency response function (FW) measurement. The principle behind t h s analysis can be summarised as follows: Response = Properties X Input

(1)

Modal parameter extraction, sometimes also called modal analysis or modal parameter estimation, is performed by curve fitting a theoretical curve for a known system to the measured frequency response using the assumption that the system is linear. When the best curve fit is obtained, the parameters of the measured system are estimated to be the same as those of the theoretical system. This paper presents the experimental route of modal analysis performed on a compact three axis vertical milling machine with a worlung volume of 500x500x500 mm. The structure of the machine is supported by six unbolted legs on a concrete foundation. This paper discusses the theoretical background, the experimental setup, the modal parameter extraction process and the results of the exercise. The theoretical background of modal analysis is presented because a sound theoretical basis is always required for an experimental procedure to be completed successfully.

2 Theoretical background of modal parameter extraction The motion of a complex mechanical structure can be adequately described by a matrix of linear differential equations as follows:

where M = the mass matrix, C = the damping matrix, K = the stiffness matrix,

{x(t)j ix(t fi {x(t)j {F(t)j

= the

acceleration vector

= the velocity vector = the

displacement vector

= externally applied forces

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Each component of the vectors in this equation represents motion at a particular point in a particular direction characterised by up to six degrees of freedom (DOFs) motion. T h s consists of translational motion in the three orthogonal directions and rotational motion about the three orthogonal axes. To extract the modal parameters from the FRF measurement obtained from a modal testing, eqn (2) must be converted into the frequency domain using the Laplace Transform and the following equation is obtained:

where { ~ ( s )=) Laplace transform of displacement responses (F(s)) = Laplace transform of applied forces (~(0))= vector of initial displacement (~(0))= vector of initial velocity When the initial conditions are set to 0, eqn (3) will be reduced to the following:

where { ~ ( s )= ) [S 2~

+ sC + K] = System matrix

(5)

To find the system dynamic, the inverse of the System matrix, called the Transfer matrix, is used to determine the relationshp between the response of the structure and the externally applied forces. Thus, eqn (4) can then be rewritten as follows:

where { ~ ( s ) = ) Transfer matrix The Transfer matrix contains transfer functions which can be expressed as the ratio of two polynomial functions of the S-variable.Each of the transfer functions describes the complete dynamic properties of the structure between a particular input DOF and a particular response DOF. To extract the modal parameters, the STARStruct software used for the modal analysis applies the following analytical expression [3]:

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where

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pk

= eigenvalues = pole

ok

= modal damping of the k'

location in the S-planeof the kthmode mode ak = modal frequency of the kthmode rk = eigenvectors = complex residue of the k' mode ui(k) = ithDOF element of the mode shape for the kthmode uj(k) = jth DOF element of the mode shape for the kthmode

The curve-fitting algorithm of the software utilises the Rational Fraction Least Squares (RFLS) method to estimate the modal parameters of the system from the FRF measurements. The following curve-fitting model is used on the FRF measurement data in the vicinity of a modal peak to find the modal frequency (ak), modal damping (o$ and complex residue (rk= rlk+jr2$:

where AO, Al, A2 = residual function coefficients. In this modal analysis, the software uses the classical or light damping assumptions to estimate the damping of the structure. T h s means that the natural frequency of each mode, Q, is much greater than its damping, o k as shown in eqn (8), and that the real part of the mode vector, RE{uk), is much greater than its imaginary part, IM{uk), as shown in eqn (9).

3 Experimental setup The frequency response of the structure was obtained by measuring the acceleration at 156 points on the structure while applying random vibration to the structure using a TIRAvib S514 electromagnetic shaker. The input force was transmitted to the structure via a stinger attached between the shaker and a dummy cutting tool attached to the spindle. The input force was measured using a PCB 221B02 ICP piezoelectric force transducer positioned between the stinger and the dummy cutting tool. This test layout is shown in Figure 1.

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Figure 1: Modal Test Layout Three PCB 629A11 tri-axial ICP piezoelectric accelerometers were used to measure the acceleration in X-, Y- and Z- directions. The sensors were moved from one set of locations to another in order to measure the acceleration at 156 points on the structure of the machine. Magnetic bases were used to attach the accelerometers to the structure providing a useful bandwidth of around 2000 Hz. In order to satisfy the modal testing principles, the shaker was suspended on a crane to ensure that all force transmitted to the structure could be measured. Additional mass was added to the shaker to increase the amount of force going into the structure. The cutting tool point was chosen as the excitation point because most of the force would be generated at this location under normal machine operating conditions. The stinger, which is a thin metal rod, was used to direct the force only in the required direction and to prevent damage to the equipment. The acceleration and force signals were introduced to a Dataphysics SignalCalc Mobilyzer dynamic signal analyser for processing. The signal analyser software then calculated the FRF measurement values at the measured points as well as their coherence. Only measurement data with high coherence values were saved for further analysis. The software also automatically exported the data in the format used by the modal analysis package [4]. In experimental modal testing, one of the most important measurements is the driving point measurement, which is the measurement at the point where the force is applied. Its value will be used in the modal analysis step to derive the

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modal scaling for the mode shapes of the structure. The FRF and coherence data of this measurement obtained in the experiment are shown in Figure 2 and Figure 3 respectively.

Figure 2: Driving Point Frequency Response Function Measurement

Figure 3: Driving Point Measurement Coherence The signal to drive the shaker was generated by the analyser and amplified by an external TIRA BA4500 amplifier before reaching the shaker. A random signal was selected to reduce the effect of any non-linearity in the machine structure on the measurement results. Throughout the measurements, the Hanning windowing technique was used to reduce signal leakage due to the non-periodic nature of the signal.

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4 Modal analysis The modal analysis of the FRF measurement data was carried out using the STARStruct software which has several curve-fitting methods to estimate the modal parameters. The method used in the analysis was the global curve fitting method which consisted of two analysis processes: the estimation of frequency and damping parameters and the estimation of mode shape. This method has the advantage that more accurate frequency and damping values can be estimated due to the use of all FRF data and more accurate mode shapes can be produced from the more accurate frequency and damping parameters [5],[6]. All measurements, including the driving point measurement, consist of 2000 sample points over a bandwidth from 0 to 625 Hz.Due to the large amount of DC offset in the data, all data were hgh-pass filtered to remove measurement frequencies below 7.31 Hz in order to obtain accurate results from the curvefitting. In addition, the low coherence values at the low frequencies, whch were caused by the dynamic performance of the shaker, were not a problem because when a mode was detected, the coherence value increased quite significantly. Modal analysis using the STARStruct software was performed by initially defining the structure of the rnachme tool using the measurement point nodes of the modal test as shown in Figure 4. The definition created consists of six substructures: the headstock, column, table, saddle, Z-axis rail guide and bed. It is used to animate the mode shapes of the structure when the modal analysis is completed. The main step of the analysis was performed by creating an advanced curve fit file in the software to find the modal parameters of the structure using the global curve fitting method. Due to the amount of data contained in the measurement results and to get more accurate results, the analysis was split into two overlapping frequency bandwidths:

The software then created two sets of stability diagrams from all FRF measurement data on both frequency ranges to find the natural frequencies and damping of the structure. These values were determined by selecting only modes with stable and still frequencies and damping on the stability diagrams. The results of the mode selection can be seen in Table 1 and the lower frequency range stability diagram with the mode selections is shown in Figure 5. Using the natural frequencies, damping and residue values found by the software, the mode shapes of the structure were calculated. They represent the vibration of the structure at each of the resonant frequencies and the summary of the mode shapes is presented in Table 2.

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Table Saddle

Figure 4: Machine Tool Structure Definition

Table 1: Natural Frequency and Damping Results Mode No. 1 2

Frequency (Hz) 28.27 93.18

Damping (%) 3.35 4.00

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Figure 5: Stability Diagram of The Lower Frequency Range

Table 2: Mode Shapes Summary

/

Mode I Frea. (Hz) 1 28 .27 93.18 2 3 116.69 4 164.46 192.30 5 236.79 6 1 7 1 267.65 8 1 2 82.29 9 1 3 19.99 10 1 384.00 11 457.1 1 12 486.54 13 509.73 57 8.40 14

I

1 / / 1

1

Sumrnarv Bendmg of the headstock and column Bending of the headstock and c o l q rockmg of the table Bendmg of the headstock and the table Twisting of the headstock, bending of the column and the table Rocking of the headstock, column and table Rocking ofthe headstock, column and table Rockmg ofthe headstock, column and table Twisting of the headstock and column Twisting of the headstock and the column Twisting of the headstock and the column Twisting of the headstock and the base, bendmg of the column Twisting of the headstock, bending of the column and the base Roclang of the headstock, bending of the column and the base Rocking of the headstock bendmg of the column. base and table U

5 Conclusions and further work An experimental modal analysis was carried out on a vertical milling machme using the STARStruct software and fourteen vibration modes were found. The modal test on the machme tool was performed over a global frequency bandwidth as opposed to on several separately predetermined frequencies. Hence, a mathematical global curve fitting method was used to identify the modes by using specialised software. The resonant frequencies and the damping of the machine tool structure were identified using the stability diagram produced by the software. Using these values, the mode shapes were then generated to illustrate the movements of the

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430 Laser Metrology und Muchiize Performance VI structure at the resonant frequencies. The description of these mode shapes are summarised and presented in this paper. From the mode shape results, it can be seen that most of the vibration occurs in the headstock and that vibration of the structure is more prominent in the lvgher frequencies. The vibration of the headstock is most llkely caused by the limited stiffness and mass on this structural component. On the other hand, the prominent vibration at the higher frequencies is likely caused by the local effects. Further modal analysis of the structure's components is planned to fully identify the potential modes and sources of vibration in the structure of the machine. Detailed investigation into the sources of the vibration will be performed to determine the nature of the vibration including the contribution of the spindle structures and the drive systems to the generation of vibration in the whole machine structure. In depth examination will also be performed on the influence of the cutting process in the vibration of the machine. An analytical approach using the finite element modelling is planned to achieve the prediction of vibration of the structure. The modelling will use the assumptions and the results of the experimental method. Additionally, the effect of bolting down the supporting legs to the concrete foundation will be investigated with regards to the structure's modal parameters using both experimental and analyhcal methods.

Acknowledgements The authors would llke to acknowledge the EPSRC financial support. The main part of the work was carried out under the EPSRC grant GRlR13401/01 for novel metrology-based control algorithms for precision manufacture.

References [l] Richardson, H.R., Measurement and Analysis of The Dynamics of Mechanical Structures. Hewlett-Packard Conference for Automotive and Related Industries, Detroit, Michigan, 1978. [2] Ewins, D. J., Modal Testing, Theory and Practice: 2nd Ed, Research Studies Press Ltd.: Baldock, 2000. [3] Spectral Dynamics, Inc., STAR System: User Guide, Part Number 34050113, San Jose, 1994. [4] Data Physics Corporation, SignalCalc Mobilyzer Dynamic Signal Analyzer: User Guide, San Jose, 2000 [5] Richardson, M. H. & Formenti, D. L., Global Curve Fitting of Frequency Response Measurements using the Rational Fraction Polynomial Method, Proc. of the 3"I ~ n tModal . Analysis Con$, Orlando, 1985. [6] Richardson, M. H., Global Frequency and Damping Estimates from Frequency Response Measurements, Proc. of the 4Ih Int. Modal Analysis Con$, Los Angeles, 1986