Structural Dynamic Modification Studies Using Updated Finite Element Model Gupta A. K., Nakra B. C. 1 and Kundra T. K. 2 IRDE Dehradun 1 NSIT New Delhi 2 Deptt. of Mechanical Engg. IIT New Delhi ABSTRACT. The aim of the present work is to develop an updated FE model of the drilling machine using analytical & experimental results, and to use this updated model to predict the effect of Structural Dynamic Modifications (SDM) on modal properties of the machine. The model updating has been done using indirect method. sensitivity based Bayesian parameter estimation technique has been applied. Modal test has been carried out on drilling machine using random noise generator. Modal identification of the experimental FRFs has been carried out using Global method. The analytical FE Modeling of the drilling machine has been carried out using beam elements. The analytical FE results have been correlated with the experimental results and MAC has been calculated. After correlation, the analytical FE model has been updated in the light of experimental data. The discrepancy between experimental and analytical model has been considerably reduced after updating. This updated FE model has been used for computer level modifications and effects of lumped mass have been studied on updated model. The updated model is compared with dynamic test results and the result has found to be satisfactory. 1. INTRODUCTION Dynamic design, which is an area of vibration and noise engineering, has acquired considerable importance due to demands on higher performance capabilities of complex mechanical and structural system like those of machine tools, engines etc. The conventional dynamic design is basically hit and trial method in which we try to achieve desired dynamic characteristics by making several prototypes. The disadvantage of this technique is that actual design cycle takes lot of time and its not cost effective. The
modifications in the actual prototype are made iteratively until the desired dynamic characteristics are obtained. As compared to conventional dynamic design, model updating based dynamic design has helped to bring finite element model close to real systems. Accordingly the effect of structural modifications on dynamic characteristics to improve dynamic design can be predicted at the computer level, making the process highly optimum one in terms of time, energy and money. The dynamic design variable can be further optimised for desired objective function of cost, weight, power etc 2. MODAL TESTING AND IDENTIFICATION OF DRILLING MACHINE Modal testing of drilling machine has been carried out in order to obtain the experimental modal properties i.e. natural frequency, mode shape & modal damping. As shown in Fig. 1, the machine tool structure has been excited at the base using random noise generator and response has measured at various points. Fig. 1: The structure under test The inertance plot for the experimental FRF has been shown in the Fig. 2. In the plot the response has been measured at point 1 while the excitation is at point 20. The modal
identification of multiple FRFs has been carried out using GRF-M method in ICATS software. The first mode has been found to be 8.79 Hz and the second mode is 44.40 Hz. Fig. 2: Inertance plot for unmodified drilling machine 3. FINITE ELEMENT ANALYSIS OF DRILLING MACHINE The Finite element model has been made in I-DEAS software. The model has been made using beam mesh. Although the FE model has been simplified but the beam elements has rotational degree of freedom which cannot be measured experimentally. Therefore the FE model need to be reduced. Guyan [7] has given the method of reducing system matrices. The FE model has been reduced using model reduction utility in FEMtools software. Since during the modal testing we have only excited in plane therefore we are correlating those modes which are in plane. As shown in table 1, the first and fourth modes are in plane and hence are required for correlation. Table 1: FEA Natural frequencies Mode No Frequency Mode shape 1 11.2 Hz In plane 2 11.4 Hz Out of plane 3 38.7 Hz Out of plane 4 63.4 Hz In plane
4. CORRELATION OF ANALYTICAL AND EXPERIMENTAL MODEL The common result of analytical analysis and modal testing is a set of modal parameters (resonance frequencies, damping and mode shape) which characterise the linear dynamics of the structure. The first stage of any reconciliation exercise is to determine how closely the experimental and analytical models correspond. Correlation analysis is the technique to quantitatively and qualitatively examine the correspondence and differences between analytically and experimentally obtained modal parameters. Modal Assurance criterion (MAC) is defined as MAC ({ },{ φ } ) φ = 2 T * { φx } { } i φa j T * T ({ φ } { φ }){ ( φ } { φ } ) X i A j * X i X i A j A where φ X - experimental mode, φ A - analytical mode Model correlation gives a direct and objective comparison of specific dynamic properties (measured versus predicted) as well as quantify the extent to which the differences (or similarities) is present. The reduced FE model has been correlated with the experimental model and MAC has been obtained. FEMtools software [6] is being used for model correlation. The comparison of analytical and experimental natural frequency has been in table 2. j Table 2: Comparison of Natural Frequency (FEA Vs EMA) Mode No FEA Frequency EMA Frequency Error (%) MAC 1 1 11.20 Hz 1 8.79 Hz -27.32 94.6 2 4 63.37 Hz 2 44.40 Hz -42.73 90.6 5. MODEL UPDATING USING INDIRECT METHOD The model updating of drilling machine has been carried out using modal sensitivity method. Modal sensitivity for the selected parameters has been found out using FEMtools software. This has been further used for model updating. FEMtools software uses Bayesian parameter estimation [6] which is a iterative method of model updating and it is sensitivity based parameter estimation. The objective of model updating is to adjust the values of selected parameters such that a reference correlation coefficient is minimised.
The FEMtools software computes several correlation coefficients. They are either based on the errors of the individual modal parameters selected as responses (resonance frequencies, modal displacements), global correlation information (MAC) or other response data like expected relative error C ri on the response value. 5.1 Model Updating Results The FE model has been updated using sensitivity method. The updating parameter selected are Young s modulus of elasticity (D) Mass density ( RHO) Cross section area (A x ) Bending moment of inertia (I Y ) The normalised sensitivity has been calculated for the above parameters for both the modes using software. With the updating parameters, the model updating programme has been executed for five iteration, with a maximum of 1% variation in the the weighted absolute relative difference (CCABSOLUTE) between resonance frequencies Table 3: Comparison of natural frequency (after updating) Mode No FEA Frequency EMA Frequency Error (%) MAC 1 1 8.79 1 8.797 Hz -0.09 94.7 2 4 44.43 2 44.40 0.06 90.3 The results after updating have been tabulated in table 3. It can be seen that error between analytical and experimental natural frequency has reduced to less than 1% after updating. 6. STRUCTURAL DYNAMIC MODIFICATION USING UPDATED FINITE ELEMENT MODEL Structural Dynamic Modification techniques are methods by which dynamic behaviour of the structure can be improved by adding the modifications like those of lumped mass, spring, damper etc. Structural modification is an area of study that deals with the effect of physical parameter changes on the dynamic properties of a structural system. These physical parameters are related to mass, stiffness and damping properties
of a system or to a combination of them. Structural dynamic modification has been done here in order to see the effect of modification on updated finite element model. The FEMtools software has a set of modification elements e.g. mass, spring, bar, beam, truss etc. Here the updated model has been taken for modification and the effect has been seen for mass modification. 6.1 Effect of Mass Modification on Updated Model To see the effect of mass modification, a mass of 20 Kg has been added on node 20 on top of the central column of the machine and the effect of this modification has been analysed. The first natural frequency has been reduced to 8.3 Hz from 8.8 Hz while the second natural frequency has been reduced to 42.8 Hz from 44.4 Hz. The mass has been varied from 20 to 100 kg and the modified natural frequency is listed in table 4 Table 4: Effect of mass modification Mass Modification (Kg) First Mode 20 8.3 42.8 30 7.9 41.6 60 7.5 40.7 80 7.2 39.9 100 6.9 39.4 Second Mode 6.2 Validation of Updated Model for Mass Modification The drilling machine structure has been modified by adding an additional mass of 14.3 Kg has been added on top of the vertical column. The result of dynamic test and updated model prediction has been presented in table 5. Table 5: Measured Frequency Vs Updated model predictions Mode No Measured Updated Model % Error frequency Predictions 1 8.37 8.40-0.35% 2 46.05 43.2 6.18%
The comparison of dynamic test and updated model prediction for mass modification of 14.3 Kg has been shown in table 6. The error for the first mode has been less than 1% while the error is more in the higher mode. The results suggest that the structural dynamic modification using updated model can be used for prediction of dynamic characteristics. 7. SUMMARY The present work has started with modal testing of a drilling machine. The experiment has been conducted on drilling machine using random exciter. The drilling machine has been excited at one point and the response has been taken at different points on the drilling machine. The frequency response function has been stored in Real time FFT analyser and is transferred to the computer through serial port communication ((RS 232C). Modal identification has been carried out in I-CATS software and modal properties namely natural frequencies, mode shapes and modal damping has been found out. The FE model of the drilling machine has been made in I-DEAS software and it has been analysed for normal mode analysis to find out analytical natural frequencies and mode shapes. Significant discrepancy has been found between analytical FE data and experimental modal analysis (EMA) data. The model has been updated and changes in the different parameters have been found out. After the model updating the updated model has come very close to actual test data and discrepancy has been considerably reduced. The updated model has been used for structural dynamic modification and has been validated experimentally. REFERENCES 1. Allemang, R.L.,and Brown, D.L., A Correlation coefficient for modal vector analysis, Proceeding of 1 st IMAC, pp. 110-116, 1982. 2. Agarwal, A., Dynamic Design of a Machine Tool Structure using Modal Testing, Simulation and SDM, M Tech Thesis, Deptt. of Mechanical Engg., IIT Delhi,2000. 3. Brown, D.L., Allemang, R.J., Mergeay, M., Parameter Estimation Technique For Modal Analysis, SAE Transaction, 88, 828-846, 1979. 4. Ewins, D.J., Modal Testing: Theory and Practice, John Willey and Sons, New York, 1984.
5. Friswell, M.I.,and Mottershead, J.E., Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, 1995. 6. FEMtools theoretical Manual Version 2.0, Dynamic design solutions, 2000. 7. Guyan, R.J., Reduction of Stiffness and Mass Matrices, AIAA Journal, 3, 2, 380, 1965. 8. Goyder, H.G.D., Methods and Application of Structural Modeling from Measured Structural Frequency Response Data, journal of sound and vibration, 68, 2, 209-230 1980. 9. Imergun, M.,and Visser, W.J., A Review of Model Updating techniques, Journal of Sound and vibration, ASME, 9-20, January 1991. 10. Kundra, T.K., Studies in identification and modification of dynamic Mechanical system, Ph.D. Thesis, Deptt. of Mechanical Engg., IIT Delhi, 1986. 11. Kundra, T.K.,and Nakra, B.C., Structural dynamic modification and model updating to improve the dynamic characteristics of machine tools, IITD-IRD Ref. RP 1085, 2000. 12. Kumar, D., Optimal Dynamic Design Studies of a Machine Tool Structure using Structural Dynamic Modification, M Tech Thesis, Deptt. of Mechanical Engg., IIT Delhi, 2000. 13. Lin, R.M., Lim, M.K., Du, H., Improved Inverse Eigensensitivity method for structural analytical Model Updating, Journal of Vibration and Acoustics, ASME, 117, 192-198, April 1995. 14. Minas, C.,and Inman, D.J., Matching Finite elements model to modal data, Journal of Vibration and Acoustics, ASME, 112, 84-91, January 1990. 15. Mottershead, J.E.,and Friswell, M.I., Model Updating in Structural Dynamics : A Survey, Journal of sound and vibration, 167, 2, 347-375, 1993. 16. Maia, N.M.M.,and Silva, J.M.M., Theoretical and Experimental Modal Analysis, John Willey & Sons, New York, 1997.