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

1. http://www.iaeme.com/IJMET/index.asp 135 editor@iaeme.com
International Journal of Mechanical Engineering and Technology (IJMET)
Volume 7, Issue 2, March-April 2016, pp. 135–147, Article ID: IJMET_07_02_016
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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

2. S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat
http://www.iaeme.com/IJMET/index.asp 136 editor@iaeme.com
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 105
MPa, ρ=7.800 tonnes/mm3
). 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.

3. Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element
Analysis Results.
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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
= 6.34 mm Where, P = 1 N, l = 100 mm, E = 2.1 x 105
N/mm2
and, the moment of
inertia Izz = 0.25 mm4
Exact Bending Stress
= 200 MPa
Where, M = 100 N.mm, y=0.5 mm.
1mm
3mm

4. S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat
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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
c1= 3.5156, c2=22.03, c3= 61.70, c4 = 120.89 and c5= 199.826
Consideration of each Moment of inertia i.e. IZZ and IYY 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 105
N/mm2,
L= 100 mm,ρ = 7.8 e-09 tonne/mm3.
.
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.

5. Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element
Analysis Results.
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Figure 1 BEAM Model with 8 elements
Figure 2 TRIA Model
Figure 3 QUAD Model

6. S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat
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Figure 4 TETRA Model
Figure 5 PENTA Model
Figure 6 HEXA8 Model

7. Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element
Analysis Results.
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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

8. S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat
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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

9. Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element
Analysis Results.
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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

10. S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat
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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

11. Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element
Analysis Results.
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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
Table 6 Consistent Mass Matrix Results for 1-D and 2-D elements
Mode
No.
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

12. S. S. Deshpande, N. P. Bandewar, M. Y. Soman and S. R. Rawat
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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.

13. Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element
Analysis Results.
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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 co-
relation 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.
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