# Application of Haar Wavelet Discretization Method for Free Vibration Analysis of Rectangular Plate Text Only Version

#### Application of Haar Wavelet Discretization Method for Free Vibration Analysis of Rectangular Plate

Kwanghun Kim1, Hyeyong Pak2

1,2department of Engineering Machine, Pyongyang University of Mechanical Engineering, Pyongyang, Democratic Peoples Republic of Korea

Suchol O3

3Institute of Science, Chongjin Mine & Metal University,

Chongjin, Democratic Peoples Republic of Korea

Chol Nam4

4Department of Resource Development Machine, Pyongyang University of Mechanical Engineering, Pyongyang, Democratic Peoples Republic of Korea

Jonggil K O5

5Department Of Machine Manufacturing, Pyongyang University of Mechanical Engineering, Pyongyang, Democratic Peoples Republic of Korea

AbstractThis paper focuses on the Haar wavelet discretization method (HWDM) for solving the problem of free vibration behaviour of rectangular plate. The displacements are expressed as Haar wavelet series and their integral. Since the constants generated during the integration process are decided as the setting of boundary conditions, therefore, the equations of motion of the entire system including boundary conditions are expressed as a series of algebraic equations. By solving the eigenvalue problem of these algebraic equation, the natural frequencies of the rectangular plate can be obtained. The accuracy and convergence of HWDM is verified against the results of the previous data, and the comparison results agree well. The effects of several geometric dimensions on the natural frequencies of rectangular plate under various boundary conditions are investigated. The numerical results for rectangular plate obtained by Haar wavelet discretization method may be served as benchmark solutions for future research.

KeywordsRectangular plate; Haar wavelet discretization method; Free vibration; Numerical analysis; Eigenvalue problem

1. INTRODUCTION

Plates are widely used in various engineering fields, and specially, in the case of ship and ocean structures they can be considered as one of the fundamental structural elements. It is very important to investigate the vibration behavior of plates which are widely used in various engineering fields. A lot of scholars have published a number of papers on the free vibration of rectangular plates [1-13]. Different methods such as Rayleigh-Ritz method, Galerkin method, finite element method and the method of separation of variables have been used to analyze the free vibration of rectangular plate. The results of the review show that although many studies have been conducted to analyze the free vibrations of rectangular plates, finding a reasonable approach to obtain the natural frequencies of rectangular plates is still an important problem. The Haar wavelet discretization method is a very simple and powerful method to solve the eigenvalue problem and the application in engineering of this method will be reviewed below.

In 1909, A. Haar proposed the Haar function, which made a great contribution to the emergence of wavelet theory. In 1981, J. Morlet  proposed the wavelet concept, which laid a good foundation for the formation of wavelet theory. Since then, the research of wavelet theory has entered a stage of rapid development. In 1981, Stromberg improved on the basis of the Haar function and found the first orthogonal wavelet. In 1982, expert Marr formed the "Mexican Hat" wavelet. In 1985, Meyer obtained a smooth wavelet orthonormal basis with a certain attenuation, namely Meyer base, which laid a good foundation for the further development of wavelet theory. In 1988, wavelet analyst Daubechies[16, 17] constructed an iterative method to construct a wavelet base that is non-zero only in a finite region (ie Daubechies base) and gave 10 presentations at a wavelet conference held in the United States. It caused a sensation, and then the book "Small Waves Ten" became a great book with great influence in contemporary mathematics, which pushed the development and practical application of wavelet theory to a climax. In 1989, Mallat[18, 19] proposed the famous "Mallat algorithm", which opened the development space for the engineering application of wavelet theory. After the 1990s, wavelet theory gradually matured. At present, research on wavelet theory has penetrated a wide variety of fields. Majak et al. [20, 21] developed Haar wavelet-based discretization method for solving differential equations, discussed both, strong and weak formulations based apporaches, and introduced this method to solve solid mechanics problems. Recently, Hein and Feklistova[22, 23] applied HWDM for solving the vibration problems of functionally graded beams under some boundary conditions.

As can be seen from the literature review, the Haar method was used for vibration analysis of various types of structures, but two-dimensional development is difficult, so there are very few examples applied to the vibration of plate structures. Therefore, the main purpose of this paper is to establish a solution method and system to apply conveniently and efficiently the Haar wavelet discretization method to the free vibration of a rectangular plate.

2. APPLICATION OF HWDM

f = f (1) f (2)

f (n )T

Haar wavelet which is a group of square waves that has

a= a a a T ,

size of +1 and -1 in some intervals and has zeros in elsewhere is one of the simplest compactly supported orthogonal

1 2 n

d 3 f (0) d 2 f (0)

df (0) T

wavelet among the wavelet families. The details of Haar

wavelet series and their integrals can be found in the following references [25-28].

b=

d 3

d d

f (0)

3 3

HWDM are used to discretize the derivatives in entire governing equations including boundary conditions. Since the

p4,1(1)

p4,2 (1)

p4,n (1)

1 1

6 2

1 1

3 3

Haar wavelet series is defined in the interval [0, 1], firstly, in

P p4,1(2 )

p4,2 (2 )

p4,n (2 )

2 2

2 1

order to apply the HWDM, a linear transformation statute 1

6 2

may be introduced for coordinate conversion from length

interval [0, L]of the rectangular plate to the interval[0,1]of the

3 2

Haar wavelet series, that is,

p4,1(n )

p4,2 (n )

p4,n (n )

n n n 1 6 2

In the HWDM, highest order derivatives of the displacements are expressed by Haar wavelet series, the lower order derivatives can be obtained by integrating Haar wavelet series.

The transverse vibration differential equation of the plate is expressed as

In practical process applications, the clamped boundary condition, simply-supported boundary condition and the free boundary condition are widely used, these boundary conditions can be written as follows in equation form.

 x x1 x x1 , = y y1 y y1x x L y y L2 1 x 2 1 y (1)
 x x1 x x1 , = y y1 y y1x x L y y L2 1 x 2 1 y (1)

 w( ) 0, dw( ) 0, 0, 1 ,dw() 0, dw() 0, 0, 1d (7)
 w( ) 0, dw( ) 0, 0, 1 ,dw() 0, dw() 0, 0, 1d (7)

Clamped boundary condition

4 w

4 w

4 w

4

4

x4

 d 2w( )w( ) 0, 0, 0, 1 ,d 2d 2w()w() 0, 2 0, 0, 1d (8)
 d 2w( )w( ) 0, 0, 0, 1 ,d 2d 2w()w() 0, 2 0, 0, 1d (8)

h

2 x2 y2

y4

k w 0

(2)

Simply-supported boundary condition

where k 4

2 , D is the flexural stiffness of the plate,

D

1

1

is defined as D E p / 12 1 2 .

Substituting equation (2) into equation (1), the vibration equation of the plate at local coordinates is defined as:

 d 2w( ) d 3w( )0, 0, 0, 1d 2 d 3d 2w() d 3w()0, 0, 0, 1d 2 d3 (9)
 d 2w( ) d 3w( )0, 0, 0, 1d 2 d 3d 2w() d 3w()0, 0, 0, 1d 2 d3 (9)

Free boundary condition

1 2 w

1 1 4 w

1 2 w

L4 2

2 L4 L4 2 2

k 4 w 0

L4 2

(3)

x x y y

 d 4 f ( ) 2 mahd 4 i1 i i (4)
 d 4 f ( ) 2 mahd 4 i1 i i (4)

By taking n=2m, f()=w(, ) (when is fixed, w is a function containing only the variable ), and the highest order derivative can be approximated by Haar wavelet along parallel to the axis:

By integrating the Eq. (4), the following derivative terms can be obtained.

Four boundary condition equations can be obtained by introducing the boundary condition, and can be written as follows in matrix form:

 f = P ab 2 b (10)
 d 3 f ( ) 2m d 3 f (0)3 ai p1,i 3d i1 d (5,a) d 2 f ( ) 2m d 3 f (0) d 2 f (0)d 2 i 2,i d 3 d 2 a pi1 (5,b) df ( ) 2m 2 d 3 f (0) d 2 f (0) df (0) ai p3,i 3 2d i1 2 d d d (5,c) 2m 3 d 3 f (0) 2 d 2 f (0) df (0)f ( ) ai p4,i 6 d 3 2 d 2 d f (0)i1 (5,d)
 d 3 f ( ) 2m d 3 f (0)3 ai p1,i 3d i1 d (5,a) d 2 f ( ) 2m d 3 f (0) d 2 f (0)d 2 i 2,i d 3 d 2 a pi1 (5,b) df ( ) 2m 2 d 3 f (0) d 2 f (0) df (0) ai p3,i 3 2d i1 2 d d d (5,c) 2m 3 d 3 f (0) 2 d 2 f (0) df (0)f ( ) ai p4,i 6 d 3 2 d 2 d f (0)i1 (5,d)

where, for the clamped boundary condition

T

f = f (0)

f (1)

df (0)

df (1) ,

b d d

p4,1(0)

p4,2 (0)

p4,n (0) 0 0 0 1

1 1

p4,1(1) p4,2 (1) p4,n (1) 1 1

P2

6 2

p1,1(0) p1,2 (0) p1,n (0) 0 0 1 0

p1,1(1)

p1,2 (1)

p1,n (1)

1 1 1 0

2

for the simply supported boundary condition,

d 2 f (0)

d 2 f (1) T

The displacement function v() can be written in the form of a matrix as following:

 f = P a1 b (6)

where

fb = f (0)

f (1)

d 2 d 2 ,

 p4,1(0) p4,2 (0) p4,n (0) 0 0 0 1 d 2g n 1 1 2 cihi p4,1(1) p4,2 (1) p4,n (1) 1 1 d i1 (17)
 p4,1(0) p4,2 (0) p4,n (0) 0 0 0 1 d 2g n 1 1 2 cihi p4,1(1) p4,2 (1) p4,n (1) 1 1 d i1 (17)

P2

6 2

By integrating second derivative of k in turn, the following

p2,1(0) p2,2 (0) p2,n (0) 0 1 0 0

expressions can be obtained.

p2,1(1)

p2,2 (1)

p2,n (1) 1 1 0 0

dg n

dg 0

and, for the free boundary condition

d ci p1,i d ,

d 2 f (0) d 2 f (1) d 3 f (0)

d 3 f (1) T

i1

fb =

2 2 3 3 ,

n dg 0

d d

d d

g ci p2,i d

dg 0

p2,1(0)

p2,1(1)

p2,2 (0)

p2,2 (1)

p2,n (0) 0 1 0 0

p2,n (1) 1 1 0 0

i1

g() can be written in matrix form as

 p2,1(1) p2,2 (1) p2,n (1) 1 1 p ( ) p ( ) p ( ) 1 c c g 2,1 2 2,2 2 2,n 2 2 R1 d d p2,1(n ) p2,2 (n ) p2,n (n ) n 1 (18)
 p2,1(1) p2,2 (1) p2,n (1) 1 1 p ( ) p ( ) p ( ) 1 c c g 2,1 2 2,2 2 2,n 2 2 R1 d d p2,1(n ) p2,2 (n ) p2,n (n ) n 1 (18)

P2

p3,1(0) p3,2 (0) p3,n (0) 1 0 0 0

p (1) p (1) p (1) 1 0 0 0

3,1 3,2 3,n

 f P1 a a = Q1 fb P2 b b (11)
 f P1 a a = Q1 fb P2 b b (11)

By combining Eq. (6) and Eq. (10) the following equation can be obtained.

where, g = g(1)

g(2)

g(n )T

as:

From Eq. (11), an unknown coefficient matrix is defined

 a 1 f Q1b fb (12)
 a 1 f Q1b fb (12)

Therefore, the fourth order derivative of displacement f

Two boundary condition equations can be obtained by introducing the boundary condition, and can be written as follows in matrix form (in this example the boundary condition is set simply supported boundary condition, in simply supported boundary condition g()= 2w/2=0, (=0, =1)):

 g p2,1(0) p2,2 (0) p2,n (0) 0 1 c R cb p (1) p (1) p (1) 1 1 d 2 d 2,1 2,2 2,nTwhere c= c c c T , d dg(0) g (0) (19)
 g p2,1(0) p2,2 (0) p2,n (0) 0 1 c R cb p (1) p (1) p (1) 1 1 d 2 d 2,1 2,2 2,nTwhere c= c c c T , d dg(0) g (0) (19)

can be expanded into the following as:

 d 4 f 1 fH1Q1 L1 f + L2 fb d 4 fb (13)
 d 4 f 1 fH1Q1 L1 f + L2 fb d 4 fb (13)

1 2

n d

1

1

Where L1 and L2 are the first n columns and the last four columns of the atrix H1Q 1 .

 g R1 c c = Q2 gb R2 d d (20)
 g R1 c c = Q2 gb R2 d d (20)

By combining Eq. (18) and Eq. (19) the following equation can be obtained.

d 4 f

d 4 f

d 4 f

d 4 f

T

1 , 2 , , n ,

d 4

d 4

d 4

d 4

p(1)

p(1)

p (1)

p (2 )

hn (1) 0 0 0 0

hn (2 ) 0 0 0 0

as:

From Eq. (20), an unknown coefficient matrix is defined

P1

P1

 c 1 gd Q2 g b (21)
 c 1 gd Q2 g b (21)

h ( ) h ( ) h ( ) 0 0 0 0

1 n 2 n n n

By using the tensor multiplying, d4f/d4can be extend to two dimensions

 4wL I w + L I f = K w M 4 1 y 2 y x x (14)

Considering the boundary conditions, it is easy to know that f is a zero vector with a length equal to 2n, and it will be omitted. Iy is the unit matrix.

Similarly,

Therefore, the second order derivative of displacement g

can be expanded into the following as:

 p(1) p (1) hn (1) 0 0d 2 g p(2 ) p (2 ) hn (2 ) 0 0 c2d dh ( ) h ( ) h ( ) 0 0 1 n 2 n n n H Q1 g N g + N g2 2 gb 1 2 b (22)

Where N1 and N2 are the first n columns and the last two

 4w 4 Ix L1 w = K yw (15)
 4w 4 Ix L1 w = K yw (15)

1

columns of the matrix H 2Q2 .

 p2,1(1) p2,2 (1) p2,n (1) 1 1 p2,1(2 ) p2,2 (2 ) p2,n (2 ) 2 1 ml n p ( ) p ( ) p ( ) 2,1 n 2,2 n 2,n n n 1 S m1 n (23)
 p2,1(1) p2,2 (1) p2,n (1) 1 1 p2,1(2 ) p2,2 (2 ) p2,n (2 ) 2 1 ml n p ( ) p ( ) p ( ) 2,1 n 2,2 n 2,n n n 1 S m1 n (23)

The following expression will be used to calculate the Haar wavelet expression of the fourth-order mixed partial derivative of w.

 4w 2 2w 2 2w2 2 2 2 22 2 (16)

If g()= 2w/2, the second derivative of g() along the parallel axis can be approximated by Haar wavelet.

Similarly, for k()=w(, )

where, l = l(1) l(2)

T

l(n )T

dl(0) T

Table 1 lists the first four order natural frequency parameters of square thin plates with constant thickness under different boundary conditions using Haar wavelet, where

m= m1 m2 mn , n d l(0)

In a similar way to equation (19)

Lx=Ly=1m, h=0.1m and material is steel (E=210Gpa, =0.3, =7800kg/m3). It can be seen that a small number of matching points can achieve good numerical accuracy, and with the increase of the scaling factor J, the numerical results

 l p2,1(0) p2,2 (0) p2,n (0) 0 1 m S mb p2,1(1) p2,2 (1) p2,n (1) 1 1 n 2 n (24)
 l p2,1(0) p2,2 (0) p2,n (0) 0 1 m S mb p2,1(1) p2,2 (1) p2,n (1) 1 1 n 2 n (24)

24

are getting closer and closer to the literature , thus proving

By combining Eq. (23) and Eq. (24) the following equation can be obtained.

 l S1 m ml = S n Q3 n b 2 (25)

Therefore, the second order derivative of displacement l

can be expanded into the following as:

 p(1) p (1) hn (1) 0 0d 2l p(2 ) p (2 ) hn (2 ) 0 0 m2d nh ( ) h ( ) h ( ) 0 0 1 n 2 n n n H Q1 l T l + T l3 3 1 2 blb (26)

3

3

Where T1 and T2 are the first n columns and the last two columns of the matrix H3Q 1 .

From above equations, we can obtain as following expression.

 2 2w 2wN1 I y N1 I y Ix T1 Kxyw 2 2 2 (27)

In this way, we can get an expression similar to Eq. (27)

that the structure is solved discretely using Haar wavelets in both directions. The natural frequency is feasible, and the boundary conditions can be accurately applied, which proves the feasibility of the method and provides a basis for the use of this method in more complex structures.

Table 2 to 5 shows the natural frequencies of the plates for different thicknesses under several boundary conditions using presented method.

TABLE2. NATURAL FREQUENCIES OF PLATE WITH CCCC BOUNDARY

 Mode h 0.02 0.04 0.06 0.08 0.1 0.2 1 179.9086 359.8172 539.7258 1468.621 899.543 1799.086 2 367.1552 734.3104 1101.466 1468.621 1835.776 3671.552 3 367.1552 734.3104 1101.466 2166.072 1835.776 3671.552 4 541.518 1083.036 1624.554 2635.765 2707.59 5415.18 5 658.9412 1317.882 1976.824 2648.261 3294.706 6589.412 6 662.0652 1324.13 1986.196 719.6344 3310.326 6620.652

 Mode h 0.02 0.04 0.06 0.08 0.1 0.2 1 179.9086 359.8172 539.7258 1468.621 899.543 1799.086 2 367.1552 734.3104 1101.466 1468.621 1835.776 3671.552 3 367.1552 734.3104 1101.466 2166.072 1835.776 3671.552 4 541.518 1083.036 1624.554 2635.765 2707.59 5415.18 5 658.9412 1317.882 1976.824 2648.261 3294.706 6589.412 6 662.0652 1324.13 1986.196 719.6344 3310.326 6620.652

CONDITION

TABLE NO 3. NATURAL FREQUENCIES OF PLATE WITH CSCS BOUNDARY

CONDITION

for

2 2w

 Mode h 0.02 0.04 0.06 0.08 0.1 0.2 1 144.7407 289.4814 434.2221 578.9627 723.7034 1447.407 2 273.82 547.64 821.46 1095.28 1369.1 2738.2 3 346.829 693.6581 1040.487 1387.316 1734.145 3468.29 4 473.3095 946.619 1419.929 1893.238 2366.548 4733.095 5 511.7808 1023.562 1535.342 2047.123 2558.904 5117.808 6 646.5281 1293.056 1939.584 2586.113 3232.641 6465.281
 Mode h 0.02 0.04 0.06 0.08 0.1 0.2 1 144.7407 289.4814 434.2221 578.9627 723.7034 1447.407 2 273.82 547.64 821.46 1095.28 1369.1 2738.2 3 346.829 693.6581 1040.487 1387.316 1734.145 3468.29 4 473.3095 946.619 1419.929 1893.238 2366.548 4733.095 5 511.7808 1023.562 1535.342 2047.123 2558.904 5117.808 6 646.5281 1293.056 1939.584 2586.113 3232.641 6465.281

.

2 2

 Kx 2Kxy + K y z = k4 Ix I y1z (28)
 Kx 2Kxy + K y z = k4 Ix I y1z (28)

Therefore, a governing equation for an orthogonal isotropic plate expressed by Haar wavelet and its integral can be obtained as:

Therefore, the natural frequency of the plate can be easily obtained according to equation (28).

3. NUMERICAL RESULTS

In this section, new numerical data that future researchers can use as benchmarks are presented along with parameter studies.

 Boundary Condition Mode J=2 J=3 J=4 J=5 Ref.  Present Present Present Present CCCC 1 3.65939 3.65014 3.64713 3.6463 3.6467 2 7.52806 7.46223 7.44301 7.438 7.4416 3 7.52806 7.46223 7.44301 7.438 7.4416 4 11.1062 11.0136 10.9777 10.968 10.974 SSSS 1 2.00959 2.00241 2.0006 2.0002 2.00 2 5.06696 5.01684 5.00422 5.0011 5.00 3 5.06696 5.01684 5.00422 5.0011 5.00 4 8.15074 8.03836 8.00963 8.0024 8.00 CSCS 1 2.94584 2.93673 2.9342 2.9336 2.9336 2 5.60883 5.56345 5.55091 5.5477 5.5484 3 7.12473 7.05063 7.03096 7.026 7.0285 4 9.73665 9.62799 9.59499 9.5864 9.5888
 Boundary Condition Mode J=2 J=3 J=4 J=5 Ref.  Present Present Present Present CCCC 1 3.65939 3.65014 3.64713 3.6463 3.6467 2 7.52806 7.46223 7.44301 7.438 7.4416 3 7.52806 7.46223 7.44301 7.438 7.4416 4 11.1062 11.0136 10.9777 10.968 10.974 SSSS 1 2.00959 2.00241 2.0006 2.0002 2.00 2 5.06696 5.01684 5.00422 5.0011 5.00 3 5.06696 5.01684 5.00422 5.0011 5.00 4 8.15074 8.03836 8.00963 8.0024 8.00 CSCS 1 2.94584 2.93673 2.9342 2.9336 2.9336 2 5.60883 5.56345 5.55091 5.5477 5.5484 3 7.12473 7.05063 7.03096 7.026 7.0285 4 9.73665 9.62799 9.59499 9.5864 9.5888

TEBLE I. FREQUENCY PARAMETERS OF PLATE WITH VARIOUS BOUNDARY CONDITIONS =L2/h(/(E1)1/2

TABLE NO 4. NATURAL FREQUENCIES OF PLATE WITH SSSS BOUNDARY

CONDITION

td>

0.06

 Mode h 0.02 0.04 0.08 0.1 0.2 1 98.68739 197.3748 296.0622 394.7495 493.4369 986.8739 2 246.8521 493.7043 740.5564 987.4085 1234.261 2468.521 3 246.8521 493.7043 740.5564 987.4085 1234.261 2468.521 4 395.1056 790.2112 1185.317 1580.422 1975.528 3951.056 5 494.19 988.38 1482.57 1976.76 2470.95 4941.9 6 494.19 988.38 1482.57 1976.76 2470.95 4941.9

TABLE NO 5. NATURAL FREQUENCIES OF PLATE WITH CFFF BOUNDARY

CONDITION

 Mode h 0.02 0.04 0.06 0.08 0.1 0.2 1 144.5568 289.1136 433.6705 578.2273 722.7841 1445.568 2 267.354 534.708 802.0619 1069.416 1336.77 2673.54 3 346.8095 693.619 1040.429 1387.238 1734.048 3468.095 4 469.3112 938.6224 1407.934 1877.245 2346.556 4693.112 5 472.3318 944.6637 1416.996 1889.327 2361.659 4723.318 6 646.5246 1293.049 1939.574 2586.098 3232.623 6465.246

Fig. 1 shows the natural frequency change of the plate with increasing A under several boundary conditions. As shown in the Fig.1, the natural frequencies are decreased as Ly/Lx increases. In particular, it can be seen that the natural frequencies are decreased rapidly until Ly/Lx=1, then are gradually decreased.

 Fig.2 The change of natural frequencies as increasing of thickness hIV. NUMERICAL RESULTSIn this paper, a reasonable analysis method based on the Haar wavelet has been presented to obtain the natural frequencies of rectangular plate with several boundary conditions. The basic principles and formulas of the Haar wavelet collocation method are described, and its approximation to displacement functions using HWDM are discussed an in-depth. The solution process of Haar wavelet used in free vibration analysis of rectangular plate is given in detail. The efficiency and accuracy of presented method are proved for natural frequencies of rectangular plate with several boundary conditions. Some conclusions obtained through numerical examples and free vibration analysis results for the rectangular plate are presented, these data may be may be served as benchmark solutions for future research.ACKNOWLEDGMENTThe authors also gratefully acknowledge the supports from Fig.1 The change of natural frequencies as increasing of Ly/Lx Pyongyang University of Mechanical Engineering of DPRK.(a)-CCCC, (b)-CFCF, (c)-CSCS, (d)-SFSF

Lastly, Fig. 2 shows the change in natural frequency of a plate with a completely clamped boundary with increasing thickness h. It can be clearly seen from the figure that as the thickness h of the plate increases, the natural frequency also increases.

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