1

DETERMINATION OF DOMINANT MODES FREQUENCY OF DOUBLE

LAYER GRIDS USING OPTIMIZED STARTING ITERATION VECTOR

Mohamad Hadi Bagherinejad1

.

Leila Shahryari2

ABSTRACT

Among eigenproblem solving methods, the vector iteration methods are one of the

fundamental equations solving methods; however, as theoretically shown, convergence to

a specific mode is difficult in such methods. In this paper, a method is presented for

double layer grid structures in which the vector iteration methods converge to the

dominant modes. For obtaining the dominant modes, the starting iteration vector is

optimized using genetic algorithm based on the mass participation ratio. Subsequently, in

order to obtain the upper dominant modes, the GramSchmidt orthogonalization method

is used. The numerical results demonstrate the computational advantages of the proposed

methodology.

Keywords: dominant modes, double layer grids, optimized starting iteration vector,

genetic algorithm, vector iteration methods, GramSchmidt orthogonalization

1. INTRODUCTION

During two recent centuries, modal analysis has been one of the strong methods of

dynamic analysis in structure engineering [1]. In modal analysis, modes and frequencies

are useful characteristics for identification of structures dynamic behavior. The

percentage of mass participation of modes shows importance of modes in the seismic

1 Post Graduate, Department of Civil Engineering, Shahid Bahonar University of Kerman, Kerman, Iran, Email:

mhb.civil@gmail.com, 2 Assistant Professor, Department of Civil Engineering, Science and Research Branch, Islamic Azad University, Fars, Iran, Email:

lshahryari@just.ac.ir.

2

responses of structures. The dominant modes have the most participation in structural

seismic responses because their mass participation ratios are more than the other modes.

The dominant modes in space-framed structures appear in primary modes, but in space

structures, the dominant modes are dispersal and the existence of dominant modes in the

high modes are possible. Therefore, developing a method to solve eigenproblems of

space structures and determine the dominant modes without calculation all modes can be

very useful also improved for all kind of structures.

All eigenvalue solving methods have iterative nature because solving eigenvalue

problems are equivalent to determine the polynomial roots [2]. The inverse iteration

method is one of the fundamental and useful methods to solve eigenproblems. Selecting a

complete unit vector as the starting iteration vector is the most important point in this

method. In solving eigenproblems, it is difficult to ensure convergence to a specific

eigenvalue; in fact, theory has shown it is impossible [2].

Many researches and studies are reported about determining frequencies and modes of

structures that some of them are mentioned here. Lanczos (1950) [*] presented a

systematic method for finding the latent roots and the principal axes of a matrix, without

reducing the order of the matrix. Wasfy and Noor (1998) [3] approximated the

frequencies, modes and dynamical responses of space trusses using fuzzy logic. Gao

(2006) [4] evaluated the frequencies and modes of truss structures using the interval

factor method (IFM) with the interval parameters. Chrysanthakopoulos et al (2006) [5]

proposed a formulae in order to approximate the three natural frequencies of two-

dimensional steel frames. Gao (2007) [6] approximated the frequencies and modes of

truss structures by combining the random factor method (RFM) and interval factor

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method (IFM) with uncertain parameters. Kaveh and Fazeli (2012) [7] obtained the

natural frequencies and mode shapes of the modified regular systems using a numerical

method based on single iteration methods. However, there is no study on developing a

method to obtain the dominant modes directly.

In this paper, the convergence moves toward the dominant modes using the optimized

starting iteration vector in order to solve the eigenvalue problem of double layer grids. In

addition, it is shown that the presented optimized iteration vector can be used for all kinds

of double layer grids. Therefore, the starting iteration vector is optimized based on the

mass participation ratio. Regarding that, each node in the space structures has three

degrees of freedom, the obtained optimized starting iteration vector can be used for

similar structures. Then, it is shown that other dominant modes can be obtained by the

orthogonalization method. Consequently, only the eigenvalues and eigenvectors of the

dominant modes are obtained without calculating all eigenvalues and eigenvectors.

2. THE METHODOLOGY

In this section, a general method is proposed to optimize the starting iteration vector.

Genetic algorithm is used for optimization. The starting iteration vector is optimized

based on the mass participation ratio. Therefore, the starting iteration vector is selected as

the variable and the mass participation ratio as the objective function of optimization

problem. For optimization by genetic algorithm, three variables are defined. Regarding

that the connections in space structures are assumed as pin or simple connection (3

degrees of freedom), the algorithm genetic variables are the elements of a 3-part vector

(V):

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(1) 1 2 3[ ; ; ]V v v v

the starting iteration vector is defined as:

(2) 1 1 2 3 1 2 3 1 2 3[ ; ; , ; ; ,..., ; ; ]Tx v v v v v v v v v

in fact, the starting iteration vector includes the 3-part vectors based on the number of

structure nodes. Then, the modes and frequencies of structures are obtained using the

inverse iteration method as below:

(3) 1 1y Mx

(4) 1 , ( 1,2,...)k kK x y k

(5) 1 1k ky Mx

(6) 111 1

( )T

k kk T T

k k

x yx

x y

(7) 1

1 0.5

1 1

kk

T T

k k

yy

x y

in which if 1 1 0Ty

(8) 1 1 as kk i k iy M and x

the participation factor ( nf ) for every mode is obtained by:

(9) Tn nf M

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in which is the matrix of nth mode and is the support efficacy matrix to show the

effect of support displacements on the structure displacements which is defined by x, y

and z. The vertical mass participations ratio (znr ) obtained by:

(10) 2

znzn

z

fr

m

in which, mz is the sum of mass structure along z-axis. As it is explained, the target is

maximization of the mass participation ratio. Therefore, the objective function for the

genetic algorithm is defined as:

(11) znObjective Function 1 r

and constraint function is defined as:

(12) 1 2 3 0Constraint Function v v v

after calculating the objective function, the entry variables are reformed based on the

objective function and the steps are repeated again. The iteration continues until the

objective function becomes minimum (zero). The only condition is avoidance of selecting

zero for each three variables simultaneously, because it leads to vagueness of the

relationships. Therefore, the optimized starting iteration vector is produced.

In order to obtain the higher modes, the Gram-Schmidt orthogonalization method is used.

In this method 1x is calculated using

(13) 1 11

m

i i

i

x x

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in which coefficients i are obtained using the conditions that 1 0

T

i Mx i 1...m and

T

i j ijM as below:

(14) 1Ti i Mx

now 1x is selected as the starting iteration vector, which is used instead of 1x and

provided that 1 0T

mx M and convergence is occurred to 1m and 1m .

3. DETERMINING OPTIMIZED STARTING ITERATION VECTOR

In this section, the elements of the optimized starting iteration vector are determined

based on the mentioned method and its performance is scrutinized. Primarily, the inverse

iteration method is used for determining the first frequency and mode of a double layer

grid structure. Then, the genetic algorithm determines the elements of the starting

iteration vector. The higher frequencies and modes are also determined using the Gram-

Schmidt orthogonalization method. The Rayleigh quotient iteration method is used to

improve the rate of convergence. In order to simplify the investigation of the presented

method, a small space structure is selected.

The geometrical characteristics of the space structure are depicted in Fig. 1. The height

between two layers is 180cm and the imposed load on the upper nodes is 10ton. The

module of elasticity is 6 22 10E kg cm , density of steel is 37850kg m and

distance between supports is 3m. The structure had 13 nodes and 32 members. The

connections are assumed as pin. Profile TUBE-D 3866.3 (Table 1) is used for all

elements. The eigenvalues, frequencies and mass participations ratios are vertically

provided in Table 2.

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Table 2 shows that the space structure has 27 modes in which the first dominant mode

occurred in the 7th mode and the other dominant modes are appeared in the higher modes.

The results of the inverse iteration method using the unit starting iteration vector are

provided in Fig. 2. As shown in the figure, the convergence is occurred to the second

mode after 14 iterations.

According to the presented method, the conditions and objective function are determined

in the genetic algorithm and the variables are defined subsequently. The population size

is 20, the number of generations is 100 and the bound for each variable is [0 1]. The

optimization procedure for the first try is shown in Fig. 3. As shown in this figure, the

objective function value is converged to 0.6168 after 100 generations. The variables

related to the minimum value of the objective function is [0 0.33]. The optimization

results are provided by five optimization tests as below:

1

2

3

4

0 0 0.33 11254.908 0.3832 targetfuncton 0.6168

0 0 0.42 11254.908 0.3832 targetfuncton 0.6168

0 0 0.86 11254.908 0.3832 targetfuncton 0.6168

0 0 0.89 11254.908 0.3832

zn

zn

zn

zn

V r

V r

V r

V r

5

targetfuncton 0.6168

0 0 0.35 11254.908 0.3832 targetfuncton 0.6168znV r

The results indicate that if the vector is selected as [0 0 a] the convergence of inverse

iteration method leads to the eigenvalue and eigenvector, which are related to the first

dominant mode.

The steps of convergence using the vector [0 0 1] in the inverse iteration are presented in

Fig. 2. As shown in this figure, the convergence is occurred to the first dominant mode

using the optimized starting iteration vector. Afterward, the proposed eigenproblem of

the space structure is surveyed. It is shown that the optimized starting iteration vector can

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obtain all the dominant modes. The determination of eigenvalues and eigenvectors

continues until the mass participation ratio reaches 0.99. The steps of convergence for

each dominant mode are presented in Fig. 4.

Fig. 4 shows that the corresponding eigenvalues of the dominant modes can be obtained

using the Rayleigh quotient iteration, Gram-Schmidt orthogonalization and optimized

starting iteration vector without determining all eigenvalues.

4. NUMERICAL EXAMPLES

Finally, several examples are proposed for scrutinizing performance of the optimized

starting iteration vector. The vector [0 0 1] is used in the starting iteration vector for all

examples. The sum of mass participation ratios must reach 0.9. The module of elasticity,

density of steel and profile of members is corresponded to the small space structure.

Table 3 shows the geometric characteristics and loading data of the examples. Fig. 5

shows the plan of the examples and Table 4 shows the obtained eigenvalues for the

examples using the optimized starting iteration vector. The table shows that the method

obtains only the dominant modes and the method stops when the sum of participation

mass ratios reaches to 0.9. In some of examples, the method does not obtain the dominant

modes according to the number of modes, which means that the upper dominant modes

might be obtained earlier than the lower dominant modes.

5. CONCLUSIONS

In this paper, in order to obtain the dominant modes in double layer grids the starting

iteration vector is optimized by the genetic algorithm. Then, by optimized starting

iteration vector, the convergence in the inverse iteration method is moved to the first

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dominant mode. Using the Rayleigh quotient iteration method with the Gram-Schmidt

orthogonalization method, the convergence is moved to the higher dominant modes by

the optimized starting iteration vector (x1= [0 0 1 0 0 1 0 0 1]). The optimized

starting iteration vector is used in several examples. In all examples, it is shown that the

optimized starting iteration vector can determine the eigenvalues and eigenvectors of

the dominant modes, without calculating all eigenvalues.

6. REFERENCES

[1] J. He, Z.F. Fu, Modal Analysis, Butterworth-Heinemann, England, 2001, pp. 1-

10.

[2] K.J. Bathe, Finite Element Procedures, Prentice Hall, USA, 1996, pp. 887-912.

[3] Lanczos, C. An Iteration Method for the Solution of The Eigenvalue Problem of

Linear Differential and Integral Operators. United States Governm. Press Office.

[4] T.M. Wasfy, A.K. Noor, Application of Fuzzy Sets to Transient Analysis of

Space Structures, Finite Elements in Analysis and Design, 29 (1998) 153-171.

[5] W. Gao, Interval Natural Frequency and Mode Shape Analysis for Truss

Structures with Interval Parameters, Finite Elements in Analysis and Design. 42

(2006) 471-477.

[6] C. Chrysanthakopoulos, N. Bazeos, D.E. Beskos, Approximate Formulae for

Natural Periods of Plane Steel Frames, Journal of Constructional Steel Research.

62 (2006) 592604.

[7] W. Gao, Natural Frequency and Mode Shape Analysis of Structures with

Uncertainty, Mechanical Systems and Signal Processing. 21 (2007) 2439.

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[8] A. Kaveh, H. Fazeli, Free Vibration Analysis of Locally Modified Regular

Structures Using Shifted Inverse Iteration Method, Computers & Structures. 107

(2012) 7582.

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TABLES AND FIGURES

List of Tables:

Table 1 properties of TUBE-D386*63.

Table 2 Eigenvalues, Frequency and mass participation.

Table 3 Geometric characteristics and loading information of examples.

Table 4 Eigenvalues obtained by optimized starting iteration vector for examples.

List of Figures:

Fig. 1 Space structure plan.

Fig 2 Results of invers iteration with unit and optimized starting vector.

Fig 3 The amounts of target function in the first optimization.

Fig. 4 Results of Rayleigh quotient iteration, Gram-Schmidt orthogonalization by using

optimized starting iteration vector.

Fig. 5 Plans of examples.

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Table 1 properties of TUBE-D386*63. d

t

d t Area J I

cm cm cm2 cm4 cm4

36.8 0.63 71.59 23410 11710

Table 2 Eigenvalues, Frequency and mass participation.

Modes number Eigenvalues 2 2/ secrad / secrad mass participation ratio 1 3644.711824 60.37144875 1.71E-31

2 4716.525831 68.67696725 6.87E-30

3 4716.525831 68.67696725 1.80E-34

4 8279.614675 90.99238801 1.54E-30

5 8333.821947 91.28976913 8.05E-31

6 8333.821947 91.28976913 1.63E-32

7 11254.90784 106.0891504 0.382614071

8 19331.87296 139.0391059 0.054988739

9 19331.87296 139.0391059 0.265001413

10 22687.02773 150.6221356 9.30E-33

11 22687.02773 150.6221356 8.87E-31

12 44050.95413 209.8831916 8.34E-32

13 45676.68294 213.72104 1.29E-33

14 66252.43191 257.3954776 8.93E-31

15 66252.43191 257.3954776 1.19E-32

16 70439.75131 265.4048819 2.87E-31

17 70439.75131 265.4048819 9.61E-33

18 79350.72903 281.6926144 0.114178681

19 83676.20901 289.2684031 0.056812749

20 94413.79436 307.2682775 0.063198709

21 94413.79436 307.2682775 0.063205639

22 155513.8388 394.3524296 5.63E-32

23 160881.2654 401.1000691 1.41E-32

24 160881.2654 401.1000691 1.09E-32

25 165352.2214 406.6352437 1.02E-33

26 182159.4772 426.8014494 1.59E-32

27 182159.4772 426.8014494 3.59E-33

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Table 3 Geometric characteristics and loading information of examples

Number of

example

Number of

nodes

Number of

elements

Number of

supports

Load entered

on each upper

nodes (kg)

Height

between two

layers (m)

1 1080 4032 8 1000 2

2 961 3644 12 2000 2

3 761 2888 16 3000 2

4 983 3756 13 2000 1

5 2464 10766 6 2000 3

6 2302 9946 10 3000 2

Table 4 Eigenvalues obtained by optimized starting iteration vector for examples

Number of example Number of

Mode Eigen value 2 2/ secrad Mass participation

ratio

1

1 310.3085 0.49829

11 1934.5171 0.08540

14 3860.3384 0.19682

21 6270.5361 0.11789

48 25153.8650 0.02223

2 1 433.0023 0.43802

5 726.5844 0.41031

33 9869.6825 0.08521

3

7 6132.5530 0.76973

14 7875.9070 0.06568

38 19103.7950 0.00072

24 13693.1700 0.03935

61 33197.2010 0.02097

57 30548.6860 0.02128

4

2 42.4086 0.72723

15 603.8357 0.01412

18 737.3313 0.14083

6 164.3654 0.00726

25 1469.8683 0.02152

5 1 4.3093 0.77923

7 137.6340 0.11498

4 52.6163 0.00675

6

2 30.2422 0.33215

5 98.8333 0.19736

3 71.0867 0.17032

9 205.1004 0.05039

6 152.6298 0.08019

18 703.1228 0.02553

14 478.7974 0.00178

16 617.3050 0.00014

12 371.5005 0.01384

22 982.9483 0.03336

14

600cm

300cm

150cm

300cm

Fig. 1 Space structure plan

Fig 2 Results of invers iteration with unit and optimized starting vector

Fig 3 The amounts of target function in the first optimization

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Fig. 4 Results of Rayleigh quotient iteration, Gram-Schmidt orthogonalization by using

optimized starting iteration vector.

16

3750cm

1650cm

1650cm

3750cm

150cm

2400cm

13

00

cm

1300cm

24

00

cm

13

00

cm

1300cm

10

0cm

Example 1 Example 2

900cm

900cm

900cm

900cm

2850cm

2850cm

150cm

900cm900cm

7800cm

41

57

cm

Example 3 Example 4

52m

104m

104m

56m

104m

56m

52m

104m

104m

56m

104m

56m

39m

39m

39m

Example 5 Example 6

Fig. 5 Plans of examples