EXPERIMENTAL MODAL ANALYSIS (EMA) OF A SPINDLE BRACKET OF A MINIATURIZED MACHINE TOOL (MMT)

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India EXPERIMENTAL MODAL ANALYSIS (EMA) OF A SPINDLE BRACKET OF A MINIATURIZED MACHINE TOOL (MMT) Rajesh Babu, K. 1, Samuel, G. L. 2 1 Research Scholar, Department of Mechanical Engineering, IIT Madras, Chennai 600 036, India. Email: kadirikotar2000@yahoo.co.in 2 Associate Professor, Department of Mechanical Engineering, IIT Madras, Chennai 600 036, India. Email: samuelgl@iitm.ac.in Abstract In many cases, modal tests are conducted on individual components of complex engineering structures where interest is confined to deriving an undamped model of the structure. In the present study, we focus on conducting modal test on a spindle bracket of a Miniaturized Machine Tool (MMT), a prototype of which is available in our laboratory. The spindle bracket is a very vital component of a MMT as it acts as a bridge between the machine s spindle assembly and its vertical slide/column. Developing an undamped model for the spindle bracket with the aid of modal testing will enable us to predict the natural frequencies of the bracket so that it allows us to visualize its impact on the natural frequency of an entire assembly of a machine tool, when pressed into operation. An experiment was conducted on the specimen by an impulse hammer and three peak values from the frequency response function (FRF) were taken into account to identify its vibration parameters viz., natural frequency, damping ratio, and modal constants by the peak picking method. Furthermore, stiffness and damping matrices of a spindle bracket were also extracted from the experimental data. Keywords: Experimental modal analysis, Frequency response function, Peak picking method. 1 Introduction In today s industry, dynamic testing is being extensively incorporated into the design development, quality control, and qualification products. A wide spectrum of devices and components ranging from large missiles, reactor coolant loops, and automobiles to delicate sensors, transducers, and microprocessor modules is tested in this manner. The primary objective of testing is to determine the dynamic behaviour of the test object. This information can be used for fault diagnosis, design improvement and evaluation of the operational capability of a system (McCormick(1981)).Dynamic tests typically involve applying a measured forcing excitation along a natural direction (degree of freedom) of the system and measuring the system response at that location as well as at other critical degrees of freedom. The wide choice of forcing excitation available impulse, sine sweep, sine dwell, sine beats, narrow band random, and broadband random depend on the source of excitation. Analyzing the measured response signals (time histories) allows determination of a variety of frequency domain and time domain information. In experimental modal analysis (EMA) this information primarily consists of natural frequencies, modal damping, and mode shapes. If an adequate number of degrees of freedom is monitored during testing, it is possible to determine a complete dynamic model for the system, in the time domain, specifically, mass matrix, stiffness matrix, and damping matrix can be extracted from test data. EMA is an experimental method to obtain the dynamic characteristics of a mechanical component of a machine tool. EMA can be regarded as a Block Box or input output approach, extracting the model from input output measurements (Gao Xiangsheng et al. (2012)). 2 Dynamic analysis of a spindle bracket For purposes of dynamic analysis, a mechanical component like a spindle bracket is modeled as a discrete DOF system. The motion of significant points on a bracket is thus described by a linear second order matrix differential equation that exhibits reciprocity between the force and the response at any two arbitrary points. This relation between force and response requires experimental measurements and 806-1

EXPERIMENTAL MODAL ANALYSIS (EMA) OF A SPINDLE BRACKET OF A MINIATURIZED MACHINE TOOL (MMT) analysis procedures that are known collectively as impedance methods. The measurements are usually obtained by Fast Fourier Transformation (FFT) based two channel digital spectrum analyzers that display frequency response function (FRF) in the form of mechanical impedance(force/response) or mobility (response/force). The graphical display of the FRF allows the engineer to gain an understanding of the behaviour of a structure during external excitations(masood. M. (1984)). The use of impact excitation together with a FFT based spectrum analyzer to determine the dynamic characteristics of structure is potentially a very attractive technique. An impact gives excitation across a broad frequency range and the upper limit of this range can be tailored to suit the particular test by varying the material of the hammer head. Therefore, it is frequently possible to investigate the whole frequency range of interest in a single test. The test is also very quick to set up since the need for connecting and aligning a shaker is eliminated. These measurements, however, can be used as input to software used to process modal parameters in order to identify the modal parameters of the structure. Such parameters can be used to verify or synthesize a mathematical model of a structure. Processing software is often based on analytical curve fitting procedures, they adjust unknown parameters in a mathematical model to minimize errors between analytical and experimental results. The software then synthesizes an analytical FRF superposed on the graphical display of the experimental FRF. 2.1 Elementary theory of EMA The primary objective of experimental modal analysis is to extract natural frequencies, modal damping, mode shapes, mass matrix, stiffness matrix, and damping matrix of a dynamic system from a set of measured responses at critical degrees of freedom (DOF) for a known excitation applied at one of them at a time. Assume that the test object (spindle bracket) can be represented by DOF discrete model and its corresponding equations of motion can be written as detailed below: We start with the classical expressions for frequency domain compliances. For viscous damping we have the equation of motion (1) where,, and are stiffness, damping and mass matrices, and andare the 1 vectors of the complex harmonic amplitudes and forces. The modal matrix is made up of the eigenvectors associated with the harmonic equation for (1) (i.e., when ). The vector can then be expressed as linear superposition of eigenvectors. To say explicitly, only in the case of proportional damping the mathematical solutioncan be derived similarly to that of the undamped system, using orthogonality properties. Modal analysis can use the eigenvalues and eigenvectors to determine the vibrational response of the system to various inputs. In modal analysis, the response at one node (or given location on the structure) is measured while a sinusoidal forcing function is input at another node. Thus equation of motion for the forced vibration problem, i.e., (2) (3) (damping neglected) must be solved. The above equation will hold good for sinusoidal forcing input,, or alternatively, the Eq. (3) can be rewritten as (4) where is a matrix called the receptance and defined as (5) 0; 1, ; (6) where the subscript is the location (node) on the structure where the displacement is measured and the subscript is the location (node) where the force is applied. The condition on the right side is the requirement that a force only be applied at location and at no other location. The calculation of can be simplified as follows (Kin N Tong (1960)): (7) (8) (9) where means a diagonal matrix and the values of are natural frequencies. Therefore, ω (10) Noting that 0 1 0 (11) 0 0 1 806-2

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India ω (12) or. ω (13) In this way the values of the receptance,, can be calculated from only the eigenvalues and eigenvectors. Physically, gives the relationship between a sinusoidal force input ( ) at node, and the sinusoidal displacement at node. It is important to realize that and are scalars and not vectors. It is common practice to replace the product of two modal constants. with and it is called the modal constant.this substitution is important because experimentally the individual values of and cannot be easily determined, only the product of the two can be found,. (14) ω In addition to using nodal displacement to characterize a structure s response, as in receptance, both nodal velocity and nodal acceleration are also commonly used. The ratio of velocity to force is called mobility and the ratio of acceleration to force is called accelerance (or inertance). Eq. (14) can be easily modified to express the quantities of mobility and accelerance. Figure 1a The CAD model of a spindle bracket Basic data for Aluminum material: Young s modulus, 71 Gpaor 71000 N mm, Volume density, 2768 kg m. Figure 1b Geometric details of a spindle bracket ω (15) (16) ω Eq. (16)represents equation for accelerance without damping condition and it is a real equation. 2.2 Spindle bracket of a Miniaturized Machine Tool The physical model of a spindle bracket and mounting fixture had been fabricated in the central workshop of the institute with a view to conduct an experimental modal analysis (EMA). The CAD model of the spindle bracket is depicted in Figure 1a. Figure 1c Schematic diagram of a miniaturized machinetool idealization of the structure Figure 1b, illustrates the geometric model of a spindle bracket with dimensions.as is evident from the Figure 1c, spindle bracket is an essential component of MMT becauseit acts as a bridge between a vertical slide and the spindle assembly. 3 Experimental modal analysis (EMA) EMA covers three phases; test preparation and setup, transfer function (frequency response) measurements, and modal parameter identification. Best results are obtained if the input force as well as the structural response is measured, allowing the transfer function (response spectrum divided by the input force 806-3

EXPERIMENTAL MODAL ANALYSIS (EMA) OF A SPINDLE BRACKET OF A MINIATURIZED MACHINE TOOL (MMT) spectrum) to be determined. If the input force is assumed to be of uniform amplitude over the frequency range of interest, and is thus not measured, then by measuring the power spectrum of the structural response, it is still possible to obtain reasonable results for resonance frequencies and mode shapes, although the damping results will have large errors. The equipment required for EMA consists of the following items (LeiGuo et al. (2010)): 1. Swept frequency oscillator (or random noise generator) and power amplifier, or an impactor (instrumented hammer) (with piezoelectric load cell and charge amplifier); 2. Accelerometer and amplifier; 3. Two channel FFT spectrum analyzer and some form of interface to a personal computer; 4. Modal analysis software; 5. Personal computer and plotter. Modal testing system is shown in Figure 2. Frontier The first part of the measurement set up is an excitation mechanism that applies a force of sufficient amplitude and frequency contents to the spindle bracket. The first step in a modal analysis test is to support the structure so that it is unconstrained and is not affected significantly by its environment. The impctor (hammer) consists of hammer tip, force transducer, balancing mass and handle (see Figure 2). Typical materials for the tip are rubber, plastic and steel. In our test, spindle bracket was supported like a cantilever beam model and was excited at its free end by an impactor. The response of the spindle bracket was measured by an accelerometer which was securely fastened to the bracket under test. The accelerometer measures acceleration of a spindle bracket andoutputs the signal in the form of voltage. This voltage signal was then processed by the FFT analyzer and other software and hardware (see Figure 2).Consequently, the required FRF plots (with frequency on X axis and accelerance on Y axis) were generated and displayed on the monitor of the computer.the experiment was repeated by hitting the spindle bracket (with the aid of hammer) at one or two locations which were very close to its free end. Figure 3 illustrates the method of conducting EMA on a spindle bracket. Accelerator Computer Specimen Force sensorcharge amplifier Hammer Figure 2 System of the modal testing 3.1 FRF measurement details The measurement for experimental modal analysis (EMA) is to acquire frequency response function data from a test structure (i.e., spindle bracket). EMA is a system identification endeavour. The structure is a black box that needs to be deciphered. For our measurement, we use force input so that FRF can be derived directly from the force and response information. Theoretically, the type of force does not matter as the FRF is defined as the ratio between the response and force. The assumption that the test structure behaves linearly is essential to attaining accurate FRF measurement. Figure 3 Spindle bracket under modal test 3.2 Analysis by peak picking method The peak picking method is perhaps the simplest single degree of freedom (SDOF) method for modal analysis. It is also called the half power method. It relies on the strict compliance of the SDOF assumption. The method treats the FRF data at the vicinity of a resonance as the data from an SDOF system. The procedure of using the peak picking method is (Jimin He and Zhi Fang Fu (2001)) a) Estimating the natural frequency: The natural frequency of the mode selected for analysis is identified from the peak value of the FRF as the. b) Estimating the damping: For estimating damping the half power points at and are located first from each side of the identified peak with amplitude. The damping loss 806-4

5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India factor or damping ratio can then be estimated from the width of the resonance peak as: 4 2 2 c) Estimating the modal constant: From the SDOF model, the FRF at the peak is known to be H. And hence, the modal constant can be estimated from H. For viscous damping model, this becomes 2H. Due to its remarkable simplicity, the peak picking method can derive quick analysis results. However, it is not capable of producing accurate modal data. This method relies on the peak FRF value which is very difficult to measure accurately to estimate the natural frequency and modal constant. Damping is estimated from half power points only. No other FRF data points are used. The half power points have to be interpolated as it is unlikely that they are two of the measured data points. It is also evident that there is no mechanism for this method to deal with noise in the measured FRF data. The peak picking method is suitable only for lightly damped FRF data (as in the present case) with well separated modes and good frequency resolution. It can perform a survey study leading to a more sophisticated analysis. For the given specimen (spindle bracket), the modal test was conducted and the required plots were obtained as illustrated in Figure 4. Figure 4 FRF plot of a spindle bracket Peaks in a frequency response (FRF) plot show the resonant frequencies of the structure. The minimums show the antiresonant frequencies at which vibration is attenuated. Each peak is analyzed by assuming it as the frequency response of a SDOF system. This assumes that in the vicinity of the resonance, the FRF is dominated by that single mode. In other words, in the frequency range around the first resonance peak, it is assumed that the curve is due to the response of a damped SDOF system due to a harmonic input at and near the first natural frequency. The point of response corresponds to that value of frequency for which the magnification curve has its maximum or peak value and the phase changes 180 deg. This is how the FRF plots are analyzed in peak picking method. From the above plots, the three peak values as seen from them are: 7100 ; 6900; 10200; 9900; 12050 ; 11900; 7000 cps; 10000 cps; 12000 cps; with 2 ----- rad/sec for 1,2,3 From the above data, the modal constants can be estimated as follows (see Figure 3): H H 1200 gn/lbf; H 700 gn/lbf; H 480 gn/lbf; (Conversion factor 1 gn = g = 9.81 m sec, and 1 lbf = 0.4536 kg 9.81 m sec ). Table 1 depicts the numerical values of dynamic parameters of a spindle bracket. Table 1: Peak picking method Dynamic parameters in SI units Dynamic parameter Peak 1 Peak 2 Peak 3 Dynamic loss factor, 0.02857 0.03 0.0125 Damping ratio, 0.0143 0.015 0.00625 806-5

EXPERIMENTAL MODAL ANALYSIS (EMA) OF A SPINDLE BRACKET OF A MINIATURIZED MACHINE TOOL (MMT) Modal constant, ( 10 ) 1462 182.7685 75.2 4 Extraction of modal parameters After the modal data have been experimentally determined, the time domain model given by Eq. (1) is extracted by using the following relationships. The stiffness matrix and damping matrix can be extracted by means of the following equations (Silva. C. W. (1984)): 0 0 0 0 0 0 0 0 0 0 0 0 (17) (18) Note that the middle matrices of the above equations are diagonal. Furthermore, because matrix inversion is needed at this stage, the fact that modal matrix nonsingular is necessary to guarantee a feasible solution. By substituting the numerical data from Table 1 in the above equations, we get the corresponding stiffness and damping matrices as useful in evaluating the dynamic characteristics of a machine tool. In the present work, dynamic parameters of a spindle bracket of MMT were extracted by conducting an appropriate modal test on the specimen after its fabrication. Mathematical equivalent equations for frequency response functions (FRF) have been formulated. These equations were used in curve fitting the experimental data which enabled us to determine the damping ratios and natural frequencies of a given structure. So far as curve fitting approach is concerned, the peak picking method has been utilized in identifying the natural frequencies and modal constants of a spindle bracket and thence its stiffness and damping matrices. References Gao Xiangsheng et al. (2012), Effects of machine tool configuration on its dynamics based on orthogonal experiment method, Chinese Journal of Aeronautics, Vol. 25, pp.285 291. Jimin He and Zhi Fnag Fa (2001), Modal Analysis, Butterworth Heinemann Publications. Kin N Tang (1960), Theory of mechanical vibrations, John Wiley and Sons, Inc. Lei Guo et al. (2010), On obtaining machine tool joints stiffness by integrated modal analysis, online IEEE. Masood, M. (1984), Impedance methods for machine analysis, Shock and Vibration digest, Vol. 16, pp. 5 14. McCormick (1981), Fine tuning mechanical design, Design Engineering, Vol. 29 34, pp. 29 34. Silva, C. W. (1984), Hardware and software selection for experimental modal analysis, Shock and vibration digest, Vol. 16, pp. 3 10. 1.9345 0 0 10 0 3.9479 0 N m 0 0 5.6850 1.2566 0 0 10 0 1.885 0 N sec m 0 0 0.9425 5 Conclusions In this work, an attempt has been made to study the structural dynamics of a spindle bracket of a Miniaturized Machine Tool (MMT) by experimental modal analysis (EMA). EMA is being extensively used in industry to evaluate the dynamic performance of products during their design development, quality control, and qualification. This experimental study is 806-6