consistent and lumped mass matrices in dynamics and their impact on finite element analysis results

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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

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 105 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/currentissue.asp?JType=IJMET&VType=7&IType=2

Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element

Analysis Results.


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 10

5 N/mm

2 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



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/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



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.


Analysis Results.


Figure 1 BEAM Model with 8 elements

Figure 2 TRIA Model

Figure 3 QUAD Model



Figure 4 TETRA Model

Figure 5 PENTA Model

Figure 6 HEXA8 Model


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



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


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



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


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

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



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.


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 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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