CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE ELEMENT ANALYSIS RESULTS

International Journal of Mechanical Engineering and Technology (IJMET) Volume 7, Issue 2, March-April 2016, pp. 135 147, Article ID: IJMET_07_02_016 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=7&itype=2 Journal Impact Factor (2016): 9.2286 (Calculated by GISI) www.jifactor.com ISSN Print: 0976-6340 and ISSN Online: 0976-6359 IAEME Publication CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE ELEMENT ANALYSIS RESULTS Prof. S. S. Deshpande and S.R. Rawat Department of Mechanical Engineering, Keystone School of Engineering, Pune N.P.Bandewar Department of Mechanical Engineering, P.V.G.C.O.E,T, Pune, India M. Y. Soman Department of Mechanical Engineering, S.K.N.C.O.E, Pune, India ABSTRACT There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. the consistent / coupled mass matrix and the lumped mass matrix. Correct determination of natural frequencies is extremely important and forms the basis of any further NVH (Noise vibration and harshness) calculations and Impact or crash analysis. It has been thought by the finite element community that the consistent mass matrix should not be used as it leads to a higher computational cost and this opinion has been prevalent since 1970. We are of the opinion that in today s age where computers have become so fast we can use the consistent mass matrix on relatively coarse meshes with an advantage for better accuracy rather than going for finer meshes and lumped mass matrix. We also find that recently in MEMS simulations involving nano- technologies such as photostrictive materials have higher frequencies and here the consistent mass matrix formulation is much more beneficial FEA has been applied successfully to almost all kinds of problems, which range from statics to dynamics. Although we find lot of literature on static analysis with respect to verification and validation of result, this is not always the case with respect to dynamic problems in frequency and time domain due to complexity of physics and numerics. The main motive in this work is to consider a standard structural configuration such as a cantilever structure model having a wide range of frequencies (Hz to few MHz) and analyse calculation accuracy on a given mesh. We present here a comparison of the consistent and lumped mass matrix results are of the view that the present analysis will assist the practical CAE http://www.iaeme.com/ijmet/index.asp 135 editor@iaeme.com

S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat Community rather than a blind use of FEA software. It is expected that the users will benefit from the understanding we had from this work. Key words: Dynamics, Finite Element Method, Finite Element Analysis, Mass Matrix, Consistent Mass Matrix. Cite this Article: S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat, Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. International Journal of Mechanical Engineering and Technology, 7(2), 2016, pp. 135 147. http://www.iaeme.com/currentissue.asp?jtype=ijmet&vtype=7&itype=2 1. INTRODUCTION The finite element analysis (FEA) is the modeling of products and systems in a virtual environment, for the purpose of finding and solving potential (or existing) structural or performance issues. Finite Element Method (FEM) has been the standard workhorse or numerical technique used for structural analysis[1,2,3] as compared to the other methods such as Finite Difference method (FDM) and Boundary Element Method (BEM).and Finite Volume Method (FVM) which are widely used for solving fluid mechanics and acoustics problems[4,5,6]. Computer Aided Engineering (CAE has become now an necessary dimension of engineering complementing the other two dimensions of pure theory and experiment. FEA softwares such as ANSYS, NASTRAN and ABAQUS can be utilized in a wide range of industries. It has also become an integral part of design process. Although much has been talked on static problems and several softwares/codes are available, in literature we find less verification and validation on standard problems with respect to dynamics especially when it comes to the determining the natural frequencies and the problems elated further say acoustics and fluid structure interaction. The basic requirement of these calculations is the correct prediction of natural frequencies. The components in general can have geometric complexity and it may not always be possible to carry out experiments on large scale actual such as a full automobile and aircraft or say large process plant piping. Thus there is lot of dependence on CAE simulations involving use of Finite element analysis.the present work analyses one standard configuration of a cantilever beam and we present here the results of several elements in terms of both the consistent mass matrix and commonly used lumped mass matrix results 2. THE PROBLEM AND THE EXACT SOLUTION The problem we have taken for analysis is a cantilever beam [7] made of steel (Young s modulus = 2.1 x 10 5 MPa, ρ=7.800 tonnes/mm 3 ). The Cross-section of the beam in y-z Plane is 1 x 3 mm. The unit of density is so selected such that it forms a consistent system of units. http://www.iaeme.com/ijmet/index.asp 136 editor@iaeme.com

Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. 1mm Figure 1 A Cantilever beam with its cross Section in Y-Z plane STATIC SOLUTION We give here both the static solution (displacements and stresses) as well as the dynamic solution for calculation of natural frequencies. It has to be noted that in static analysis we solve the final algebraic set if equations [K][X]=[F] (1) And in case of dynamics we solve the following eigenvalue problem [M][ ]+[K][X]=[0] (2) In using the mass matrix, the two approaches are 1. Consistent mass matrix: This is obtained by using the shape functions [2] for the elements and is given by [ ] = (3) This involves off diagonal entries and also referred in the CAE community as full or coupled mass matrix in FEA softwares. 2. Lumped Mass Matrix: It is a diagonal matrix obtained by either row or column sum lumping schemes commonly used in literature [3].It presents a computational advantage especially in the problems of impact /crash analysis as the procedure involves then a mass matrix inversion. It is to be noted that mass doesn t play any role in static analysis and hence it is immaterial whether we use the consistent or lumped mass matrices. Exact Deflection inertia I zz = 0.25 mm 4 = 6.34 mm Where, P = 1 N, l = 100 mm, E = 2.1 x 10 5 N/mm 2 and, the moment of Exact Bending Stress = 200 MPa Where, M = 100 N.mm, y=0.5 mm. 3mm http://www.iaeme.com/ijmet/index.asp 137 editor@iaeme.com

S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat This is the standard Strength of Materials solution which can be found in any book [8]. DYNAMICS Natural Frequencies Calculation The cantilever structure is a continuous type of system and has infinite natural frequencies but we have considered first ten natural frequencies in this paper. A standard formula in the literature [9,10] is Where the first five coefficients are c 1 = 3.5156, c 2 =22.03, c 3 = 61.70, c 4 = 120.89 and c 5 = 199.826 Consideration of each Moment of inertia i.e. I ZZ and I YY gives us two frequencies and hence we can calculate the first ten natural frequencies of the structure. As a sample calculation and Where, E = 2.1 x 10 5 N/mm 2, L= 100 mm,ρ = 7.8 e-09 tonne/mm 3.. We can calculate the other natural frequencies and we present here a table of the exact solution below. TABLE 1 Exact natural frequencies in Hz Sr. No. Frequency Value in Hertz Brief Description Of the mode Shape 1 84.62 Bending in Y-Z Plane 2 253.88 Bending in X-Z Plane 3 524.25 Bending in Y-Z Plane with 1 node 4 157 Bending in X-Z Plane with 2 nodes 5 1470.9 Bending in X-Z with 1 node 6 4412.7 Bending in Y-Z with 3 nodes 7 2881.923 Torsional Mode 8 8645.77 Bending in X-Z with 2 nodes 9 4763.706 Bending in Y-Z with 4 nodes 10 14291.117 Bending in Y-Z with 5 nodes 3. FINITE ELEMENT MODELING In order to understand the difference between Consistent and lumped mass matrix, we have used 8 elements in a length of 100 mm for all the models from one dimension to three dimension. i.e. 8 beam elements were considered in one dimension, with the beam cross sectional properties such as area and moment of inertias in two planes. For shell elments two elements were used for representing the width and a thickness of 1 mm was assigned as physical property.the three dimensional representation also followed using the same logic of 8 elements along the length and split of two along the width and one split along the thickness. A typical representation of these meshes is shown in the figures given below. http://www.iaeme.com/ijmet/index.asp 138 editor@iaeme.com

Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. Figure 1 BEAM Model with 8 elements Figure 2 TRIA Model Figure 3 QUAD Model http://www.iaeme.com/ijmet/index.asp 139 editor@iaeme.com

S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat Figure 4 TETRA Model Figure 5 PENTA Model Figure 6 HEXA8 Model http://www.iaeme.com/ijmet/index.asp 140 editor@iaeme.com

Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. Figure 7 Typical static deformation plot for a quadrilateral element. Bending in y-z plane Figure 8 First mode shape. Bending in X-Z plane Figure 9 Second mode shape. Bending in X-Z plane Figure 10 Third mode shape. Bending in Y-Z plane http://www.iaeme.com/ijmet/index.asp 141 editor@iaeme.com

S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat Figure 11 Fourth mode shape. Bending in X-Zplane Figure 12 Fifth mode shape. Bending in X-Z plane Figure 13 Sixth mode shape. Bending in Y-Zplane http://www.iaeme.com/ijmet/index.asp 142 editor@iaeme.com

Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. Figure 14 Seventh mode shape Torsional mode Figure 15 Eight mode shape Bending in X-Z plane Figure 16 Nineth mode shape Bending in Y-Z plane http://www.iaeme.com/ijmet/index.asp 143 editor@iaeme.com

S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat Figure 17 Tenth mode shape Bending in Y-Z plane 4. RESULTS AND DISCUSSIONS The results for each element have been tabulated here. The runs were made by a finite element programme SADHANA (Static and Dynamic High end Analysis using Novel Algorithms) written by the first author in FORTRAN 77. The main reason of not using a commercial software was that some have the capability of using only lumped mass matrix and sometimes the option of using coupled mass is available only to limited extent only for some elements. We also give a comparison of exact solution vs. the FEA solution for each element presented so that the practical finite element user get s understood by the user. This comparison was given by the first author recently [11] but by using NASTRAN solution and the SADHANA results agree closely with that of NASTRAN. The SADHANA programme is getting updated to C++ language and writes the results compatible with current post processor HYPERVIEW and others. Table 2 Dynamic Results Using Lumped Mass Matrix for 1-D and 2-D element f= Natural Frequency (Hz) f BEAM TRIA3 TRIA6 QUAD4 QUAD8 EXACT 1 83.23 84.145 83.934 83.7761 83.996 84.62 2 249.453 518.834 256.113 251.5651 251.300 253.88 3 507.569 1152.093 527.778 516.524 525.43 524.25 4 1394.440 1437.234 1472.077 1428.398 1471.100 1470.92 5 1521.836 2790.206 1596.777 1571.495 1569.135 1572 6 2676.007 3294.584 2888.589 2767.835 2878.696 2881.92 7 4159.559 4586.063 4546.4363 3562.411 4297.599 4412.7 8 4294.024 6744.069 4595.034 4442.262 4364.554 4763.7 9 4513.327 6826.454 4798.913 4518.742 4758.308 8645.77 10 6060.620 9021.878 7238.455 6596.528 7112.984 14291.12 http://www.iaeme.com/ijmet/index.asp 144 editor@iaeme.com

Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. Table 3 Dynamic Results Using Lumped Mass Matrix for 3-D element f TETRA4 TETRA10 PENTA HEXA8 1 548.507 83.290 84.381 83.969 2 1067.631 249.452 528.95 254.439 3 3643.972 525.943 1157.875 526.859 4 5560.945 1487.258 1508.601 1494.45 5 9823.528 1568.645 3082.873 1588.43 6 13062.53 2971.311 3956.420 3062.235 7 15448.89 4230.298 5528.370 3776.604 8 19881.87 4417.615 6848.260 4498.5603 9 27522.23 5038.961 9537.609 5476.641 10 30816.87 7718.752 11908.08 9113.109 Table 4 Static Results for 1-D and 2-D element δ= Deflection (mm), σ = Bending Stress ( MPa ) Sr. No. Type of element δ σ 1 BEAM 6.345 200 2 TRIA3 6.242 176.707 3 TRIA6 6.337 204.437 4 QUAD4 6.2887 192.87 5 QUAD8 6.338 201.67 6 EXACT 6.34 200 Table 5 Static Results for 3-D elements Sr. No. Type of element δ σ 1 TETRA4 0.135 153.567 2 TETRA10 6.403 147.781 3 PENTA 6.210 113.381 4 HEXA8 6.252 194.432 Mode No. Table 6 Consistent Mass Matrix Results for 1-D and 2-D elements BEAM TRIA3 TRIA6 QUAD4 QUAD8 1 83.85 84.85 84.23 84.54 83.62 2 251.23 549.34 252.34 253.35 251.32 3 525.45 1161.43 537.45 549.56 517.23 4 1473.56 1643.34 1549.65 1643.20 1422.34 5 1572.67 3584.32 1602.45 1664.32 1574.56 6 2892.32 4645.94 3180.56 3676.78 2736.75 7 4392.23 6846.67 4549.50 4912.89 4373.50 8 4511.78 7117.45 5002.34 5112.56 4451.23 9 4802.89 12208.23 5603.32 7212.34 4603.45 10 7221.23 13102.12 9071.45 10710.360 6602.34 http://www.iaeme.com/ijmet/index.asp 145 editor@iaeme.com

S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat Table 7 Consistent Mass Matrix Results for 3-D elements Mode No. TETRA4 TETRA10 PENTA HEXA8 1 551.23 83.32 85.28 84.72 2 1072.34 249.43 530.61 256.98 3 3692.43 526.56 1169.23 551.23 4 5691.92 1492.34 1600.45 1662.40 5 10103.33 1573.67 3180.41 1663.23 6 13104.56 2892.45 3400.21 3681.12 7 16002.89 4391.12 5536.45 4951.23 8 21005.78 4826.23 6850.49 4982.30 9 29305.78 5072.34 9642.76 7252.23 10 33706.54 7762.56 11927.13 10901.56 Table 7 Relative comparison of computational cost of using consistent Mass Matrix Sr. No. Type of element CPU of CMM / CPU of LMM 1 BEAM 1.004 2 TRIA3 1.02 3 TRIA6 1.34 4 QUAD4 1.56 5 QUAD8 1.78 6 TETRA4 1.35 7 TETRA10 1.50 8 PENTA 1.45 9 HEXA8 1.76 The above parameter gives the relative cpu time for computation of natural frequencies for the element with respect to the lumped mass matrix. This parameter is computed by CPU time taken by using consistent mass matrix divided by cpu time taken by lumped mass matrix. The user can get an idea of using the consistent mass matrix for large problems by appropriate interpretation of the degrees of freedom for the model. The observations are as follows: 1. The line element representation by BEAM gives a good prediction for first 3 natural frequencies. The consistent mass matrix results are closer to exact ones for these frequencies. But a deviation is observed from fourth frequency onward. 2. The TRIA 6 and QUAD8 elements perform well upto first five frequencies but then the deviations are present from sixth natural frequency. 3. The first order triangle TRIA3 is too stiff and is able to represent only the first natural frequency properly. The deviations from exact solution are much larger as compared and even the consistent mass matrix is of no help on this. It is very difficult to give an opinion a the same element performs well for highest frequency. Overall consistent mass matrix accuracy is better than the lumped mass matrix one. 4. The same is the observation with the PENTA. Performs well on higher side bnot in the lower and mid frequency range is what one can say on the performance of this element. 5. The TRIA6 and QUAD8 perform well in low frequency regions but give lesser values as comared to exact solution. The tend of underprediction continues. 6. Most of the elements perform very well in static except TRIA3 and TETRA4 and PENTA which is a well-known fact, that these are stiff elements and predict the displacements to a low value. TRIA6 and QUAD8 over predict the stress, a fact not so well CAE known in the community 7. Oveall the consistent mass matrix values are higher than that of the lumped mass matrix. http://www.iaeme.com/ijmet/index.asp 146 editor@iaeme.com

Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. 5. CONCLUSIONS We discover that there are still lot of unanswered questions when it comes to the interpretation of results of dynamic simulations and these need to be taken care of by the practical user. Theoretical analysis in terms of calculations available on standard configurations may be helpful and also the experimental validation. Physical understanding is much more important and its correlation to the numerical with respect to element plays a very important role. We have pointed out our observations but the question of underprediction of higher frequencies and whether we shuld use elements like PENTAS, TETRA4 for higher frequencies is not yet answered and not yet addressd in finite element literature. Is it a matter of pure coincidence or something which we have not known uptill now is a question.the finite element solution is a discrete approximation where there are further complexities for practical problems as several mesh quality parameters such as distortion or Jacobian, aspect-ratio, skew or taper, min and max angles of the element come into picture. With increase in computer speed we now are of the opinion that full advantages of consistent mass matrix can be taken it is advantageous in MEMS, Nano structures with higher frequency content, mesh refinement is one solution but similar are the computational times for a practical problem hence we show that we can easily use the option of consistent mass matrix. In our view, the scope of experimental analysis is more critical and should be more encouraged. The finite element user has to keep always in his mind the corelation of a suitable mesh and experimental value pertaining a particular mode shape and should validate the finite element model accordingly for further studies on frequency response calculations. REFERENCES [1] Zeinkeinwicz, O. C., Finite Element Method, Butterworth Heineman, 6th Edition, 2005. [2] Logan, D. L.,A First course in Finite Element Method, 5th Edition, Cengage Learning, 2015. [3] Petyt, M., Introduction to finite Element vibration analysis, 2 nd edition, Cambridge Univrsiy Press, 2015. [4] Tannehill, J.C, Anderson,D.Aand Pletcher, R.H, Computational Fluid Mechanics and Heat transfer, Taylor and Francis Indian edition, 2012. [5] Chung,T.J, Computational Fluid Dynamics, Cambridge University Press, 2010 [6] Banerjee, P.K, and Butterfield, R. Boundary element Method in engineering science, McGrawHill 1981 [7] Gokhale N.S, Deshpande, S. S, Bedekar, S.V.and Thite A.N, Practical Finite Element Analysis, Finite to Infinite Publications, 2008. [8] Ramamurthy, G., Applied Finite Element Analysis, I K International Publishing House, 1nd Edition, 2010. [9] Wu, Shen, R., Lumped mass matrix in Explicit Fintite Element Method for Transient Dynamics of elasticity, Computer Methods in Applied Mechanics and Engineering, 2006, 195(44) pp 5983-5994. [10] Rao S.S, Mechanical Vibrations, Pearson India,5th Edition,2014 Bandewar N, P and Deshpande, S.S, Comparison of several elemehts and their performance in static and dynamic FEA, Paper presented at 2nd National Level conference on Advances in Mechanical Engineering Techniques, 17-18 March 2016, Pune. http://www.iaeme.com/ijmet/index.asp 147 editor@iaeme.com