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Vibration of plates with straight-line, clamped and free edges
Journal
ofSound
and Vibration (1986) W(3),
VIBRATION
437-447
OF PLATES WITH
CLAMPED
AND K.
STRAIGHT-LINE,
FREE
EDGES
POLTORAK
Institute of Machine Design Fundamentals, Warsaw Technical University, 02-524 Warszawa, ul. Narbutta 84, Poland (Received
1 August 1984, and in revised form 16 February 1985)
This paper deals with vibration problems of thin plates having straight-line, mutually perpendicular, clamped and free edges and subjected to a load consisting of a set of transverse, arbitrarily located random forces. It is assumed that the number of edges of a plate forming recurring figures is optional but each of these edges is either clamped or free along its entire length. The procedure for solving the boundary problem based upon the R-functions method and for estimation of transverse displacements based upon the correlation analysis is presented. Numerical calculations are carried out for two example plates. 1.
INTRODUCTION
Many authors have investigated vibration problems of plates with irregular boundaries. So far however only relatively simple shapes of plates have been considered because the boundary value problems of such plates with mixed boundary conditions turn out to be very complex. For instance solid plates with arbitrary outer edges and plates with doubly connected arbitrary shapes have been investigated. For these problems general approximate methods such as finite element, finite difference and point matching methods [l-3] as well as conformal mapping techniques [4,5], the constant deflection lines procedure [6] or the Fourier expansion collocation method [7,8] have been usually applied. Recently a new so-called “R-functions” method has been developed by Rvatchev [9, lo]. By means of this method any shape of plate with homogeneous or mixed boundary conditions can be treated. However in case of very complex plates it yields very complex analytical expressions which differ for each shape of the plate. So, in general, a universal computer program for arbitrarily shaped plates cannot be established. Rvatchev’s approach is based upon the Galerkin-Ritz method. However the basis functions which in linear combination approximate the solution are determined in a specific way, so that they can satisfy complex boundary conditions of arbitrarily shaped plates. In the first step of this procedure a transformation of logical expressions describing the area of the plate into analytical ones by using properties of R-functions is performed. As a result one obtains “contour functions” w(x, y) satisfying the following conditions:
[WIG=o,
w2 0 within the plate area,
w E ck+‘,
awlan
= 1.
(la-d)
Here G means the edge of the plate and n the normal direction to the edge. Afterwards the contour functions are substituted into general formulae for basis functions corresponding to a given combination of boundary conditions. The form of the contour functions significantly influences the complexity of this substitution and the following Galerkin-Ritz procedure since they are subject in this process to troublesome multiple differentiation, which cannot be performed numerically. For example, in the case of a plate with partly clamped and partly free edges the required order of differentiation is equal to 5. 437 0022-460X/86/030437+
15 $03.00/O
0
1986 Academic Press Inc. (London) Ltd.
438
K. POLTORAK
It is the purpose of this paper to show that when some restrictions on the shape of the plate are imposed then simple universal recurrence formulae for the contour functions, in a suitable form for computational analysis, can be derived. Since they are universal, the multiple differentiation has to be carried out analytically only once. So any change of the shape of the plate, within imposed assumptions, will not demand changes in the once derived expressions. Investigations have been carried out for plates having both clamped and free edges. The restricting assumptions are as follows. The plates have only straightline, mutually perpendicular edges. The number of edges forming recurring figures is optional but each of these edges is either clamped or free along its entire length. In the second part of the paper the probabilistic response of such arbitrarily shaped plates is considered. Previously the R-function method was applied in probabilistic problems by Waberski [ 11, 121 who investigated the response of plates subjected to random loading having the character of a measurable time-space field. In this paper a load consisting of a set of concentrated, transverse, arbitrarily located random forces is taken into consideration. The number of forces is unlimited. The response of the plate is found by reducing the problem to the problem of a system of many inputs and one output, and on the basis of correlation analysis. A plate of example shape subjected to an example load is shown in Figure 1. AY
Figure
1. The example
plate subjected
to the example
load.
The procedure presented in this paper can be useful in a great variety of technical problems because the restriction on the shape of the plate boundaries does not seem to be very strong if one notices that techniques applied to shape elements often produce similar restrictions. Numerical calculations have been carried out for two example plates, being simplified models of excavator platforms subjected to loads coming from engines.
2. SOLUTION
OF BOUNDARY
PROBLEM
The basis functions simultaneously satisfying boundary conditions on both clamped and free edges, regardless of the shape of the plate, have the form [9] Hij=wl
I
&w:A,- 6(wiW:'wi)[D~(w:h)+6[w:(D,Ai+A,D,w,)lA,D,(w:)I 1
+(2- Y)[~,~~(w:~~)+~w:A~~~~w~II
(2)
VIBRATION
OF
PLATES
WITH
STRAIGHT-LINE
439
EDGES
where
dg = Pi(~lX)Rj(V,y),
A,=[~/~(~:+~:)I[~,(w:~~)+~~(w:~~)I,
Pi and Rj being Tchebyshev’s polynomials of the first kind, n1 and n2 normalizing factors in accordance with plate dimensions, v2 Poisson’s ratio, and D,, D2, Dj, T, and TZ differential operators defined by 1=k
,=k
T,J=
Dkf= C
c ,=(,
i=O
iiyi(g)‘(E)k-‘. (-‘)c:dxk:k{ (3)
w,(x, y) and w,(x, y) are contour functions. The function w, is related to clamped edges and the function w2 to free edges of the plate. In order to find universal formulae for the functions w, and w2 one first considers two types of infinite areas-a vertical and a horizontal belt-in the plane co-ordinate system (see Figure 2). The areas of the belts including their boundaries can be described by simple inequalities: for the vertical belt,
R,=-ab+(a+b)x-x*20
for the horizontal belt.
LA,= -cd + (c + d)y - y2 2 0
Figure 2. The vertical
and horizontal
(4)
belts.
With i the belt number, one can define Xi = -Uibi + (ai + bi)X - X2,
K=-c,d,+(c,+dJy-y2.
(5)
Then by use of logical functions one can create an expression for a rectangular area: fl,=f2,nR,,
or
R,, = Xi 2 0 n Y, 2 0
(I means “rectangle”,
i = 1, 2 . . . n). (67)
This area will be called the elementary area. By use of Boole’s function, defined as
s,(s)={; p:;},
(8)
expression (7) can be rewritten as fl,i=[S>(Xi)=l]n[S,(
Yj)=l].
(9)
In order to transform the logical expression (9) into an analytical one one introduces the R-functions, which have the property that their signs depend only on the signs of the
440
K. POLTORAK
arguments. In this paper the following R-functions have been used: XA y=x+
Y-(x2+
Y2)1’2,
xv
y=x+
Y+(x*+
Y2)“2,
x=x.
(10)
These correspond to the logical product, sum and complement respectively. Thus, instead of (9) one can write 0,i = Sz(Xi) A S2( Y) 2 0
(II)
and, upon utilizing the properties of Boole’s functions (8), 0,i = S,[ Xi A Y] 2 0.
(12)
R,i = Xi A Y 2 0,
(13)
This yields finally
or, in the case of the complement of the rectangle including its boundaries, OCr,i= Xi A yi 2 0
(CTmeans “complement of the rectangle”).
(14)
Now it is evident that the function wi=XiA
Yi=Xi+
Yi-(X:+
Y?)“‘=O
(15)
describes the boundary of the rectangle. Naturally wi=Xi A Yi=Xi
A Yi=O.
(16)
Now one can consider the area as a product of complements of n rectangles (Figure 3). If one denotes (xY)i = xi + r, - (Xf-t Y:)“*, (XY),i = (Xf+ Yf)1’2- Xi - Y
(c means “complement”),
Figure 3. The product of complements of n rectangles.
Figure 4. The product of interiors of m rectangles.
(17)
VIBRATION
OF
PLATES
WITH
STRAIGHT-LINE
441
EDGES
then in accordance with expressions (6)-( 14) one has, for this area, n pE,n= (. . -((G,l n J&J n f&d n - - -) A fL =((...(((XY),,~On(XY),2~O)n(XY),,~O)n’..)n(XY),,sO)
**(((XVc,, A(Jmc,,) A(X%3 A*. *IAwn,,1
= a(*
(18)
= (. . .(((xy)c,, A(xy),,) A(xy),,) A. *‘1A(xy)c,n 2 0, (pc means “product of complements”). replaced by the recurrence formula =
(XV,,”
Cxy)pc,ili=n
=
The last member of expression
txy)p,i
-I+
(xy)c,l
-[(xy)‘,c,i-l
+
(18) can be
(xy)zC,i11’2~
i=l,2 ,..., n, (19) where (XY)PE.I = (XY),,. A similar procedure can be repeated for the area formed by the interiors of m rectangles with their boundaries (see Figure 4). As a result one obtains (XY),,
= (XY)p,ili=, =(XY),i-l+(XY)i+[(XY)i,i-l+(XY):]”’,
(20)
where (XY),, = (XY), (p means “product”). One can then form one more area, being a product of L$c,n and LZ,,,. Then the boundary of this area is described by the equation (XY) = (XY),,,
- [(XY)&, + (xv);,,]“’
+ (XY),,
= 0.
(21)
By making use of the function (XY) one can obtain the functions wi and w2 for any plate shaped in accordance with the general assumptions presented in the introduction to this paper. This is due to the fact that the function (XY) satisfies conditions ( la, 1b, 1c). Condition (Id) can also be fulfilled after simple modification of (XY). It can be proved that condition (lc) will be satisfied by w1 or w2 if the same condition is satisfied by all functions (XY)i: i.e., the functions describing boundaries of elementary areas used in the procedure. The modification can be easily carried out by rewriting functions (5) in the following way: Xi=[l/(bi-a,)][-aibi+(ai+bi)x-x2],
Y=[l/(ci-di)][-c;di+(ci+di)y-y*]. (22)
Having the functions Hij that satisfy boundary conditions one supposes that the solution of the equation of motion is of the form
U(-%Y, t) = f
i+j=O
Hc(X,Y) Kj(t)v
(23)
or, upon replacing the index matrix by the index vector for simplification of expressions,
UC% Yv t) = f &(X3 Y) V(t),
n=(m+1)2.
(24)
i=l
The equation of motion is transformed reference [ 111, into the set of equations
i,
{
V,D[Hi, 41 +
by means of Galerkin’s method, according to
(2pbP$+ Pb!$)}(H. Hj) = 11 .f(K
Y,
t)Hj dx dY,
R
j=1,2
3. . . n,
(25)
442
K. POLTORAK
where h is the thickness, p the mass density, p the damping factor, f(x, y, t) the applied dynamic load, D = Eh3/ 12( 1 - Y’) the tlexural rigidity, E the Young’s modulus, 0 the plate area and d2Hia2Hj
[H,, Hjl =
a2Hid2Hj
sa42+i)~2~-2
a2Hi d2Hj ax dy ax ay
n
dx dy,
HiHj dx dy,
(Hi, Hj) =
(26)
(27)
R A being the Laplacian operator. Eigenvalues can be calculated from the determinant of the symmetric matrix:
det I[Hi, HI] - A‘( Hi, Hj)I = 0.
(28)
The natural frequencies are wj = A,(D/ph)“*/A*,
(29)
where A denotes herein a chosen characteristic in-plane dimension of the plate. Equations (25) can be solved separately one by one only in the case when functions Hi satisfy orthogonality conditions: (30)
The functions Hi that satisfy condition (30) can be denoted by Gi. Actually they 1 are normalized eigenfunctions of the problem. Thus, one can obtain
& =i
j=1
avHj
I? ati%, i
j=l
where the av are coefficients to be determined equations
[LLjl{a,-I= 0,
j=l
a&
, >
from the set of homogeneous
i=l,2,...n,
(31) matrix (32)
where LLj = [ Hk, Hj] - A:( Hk, Hj). Each of these equations can be solved with accuracy to within a multiplicative constant by transformation into a non-homogeneous set of n - 1 equations, by putting air = 1 and applying Cramer’s formulae. 3. ESTIMATION OF TRANSVERSE DISPLACEMENTS By estimation of transverse displacements is meant herein the determination of displacement variances for selected points of the plate. The load, consisting of a set of transverse, arbitrarily located, random and mutually correlated forces, can be represented as
f(x, Y, t) = I, 5(t) 6(x, --xl S(Yr -Y),
(33)
where Z,(t) is a random force, s is the number of forces, S( ) is Dirac’s delta function, and x, and y, are the co-ordinates of the force application points. Then, by making use of the Dirac delta function properties the right-hand side of equation (25) can be
VIBRATION
transformed
OF
PLATES
WITH
STRAIGHT-LINE
443
EDGES
in the following way:
6(&-X)
S(Yr-Y)Hj(X,
Y)
dx
R
dY=ir<1Zr(t)Hj(Xn Yr).
(34)
Thus, the set of equations (25), upon taking into consideration e_quation (34) and replacing the functions H(x, y) by the corresponding eigenfunctions H(x, y) (see equation (31)) can be given in the form (35) Of course, now each of these equations can be solved one by one because terms other then those in which i =j, according to the orthogonality of the eigenfunctions, have vanished. The steady state solution is given in the form cc
I
G,( T)Z,( t - T) dr,
(36)
0
where Gj = (l/p,) e-+7 sin Pjr is the Green function, pj = (~3 - P~)“~ for w > )(L,and T is the time difference. Finally the displacement function can be expressed by Gj( T)Zr(
and the correlation
t
-
T) dr.
(37)
function for a chosen point P(x, y) of the plate by
K,(x, Y, 7) = & X
i$, ,il r$l “$, tTl,Cx9 Y)fiCx9
Ylfiitxn
Gi(T1)Gj(T2)RN(T-T2+T,)
Yr)fiCxw
dT1
dT2,
Yu)
(38)
where r,u=l,2 ,..., s, and R,( T - TV+ TV) are the mutual correlation functions of the random forces. The variance of the displacement can be easily obtained from formula (38) by putting T = 0. 4. NUMERICAL EXAMPLES In order to illustrate the use of the proposed procedure numerical calculations have been carried out for two example plates subjected to two variants of load. The first plate (see Figure S(a)) is rectangular and has an inner rectangular hole. The second one (see Figure 5(b)) is also rectangular but has nine inner rectangular holes. Both plates are clamped on the edge of the central hole. The remaining edges are free. These plates are simplified models of excavator platforms. In the first variant the plates are subjected to a single random force and in the second variant to a set of four forces. The dimensions of the plates and the locations of the forces are shown in Figure 5. The material data, necessary for some calculations, have been chosen as for steel. The mutual correlation
444
K. POLTORAK Y
t
-150
-50
50
150
x
-75
?? A ?? Gl
‘I, i,
i, -250 (a)
(b)
Figure 5. Two example plates investigated. of four forces. (a) Plate 1; (b) plate 2.
Dimensions
in cm. Z,, the single force;
Z,, Z,, Z,, Z,, the set
functions of the applied forces are assumed as R, = (1/2)“(u,+a,)
eC_T-p~a~,
‘=
a, and a, being the standard deviations of the random forces, LX,.,, and &-coefficients determining the degree of mutual dependence, and a,.,, the distances between the forces. These functions have been used to determine the variance of the displacement by substituting them into equation (38), executing the integration and putting r = 0. In these examples u,, a,, u3, a, = lo3 N, all coefficients (Y,,, = 15 x 103(l/s), and all- coefficients (Y, = 0.7. The following results have been obtained: (1) the relation between eigenfrequencies and the plate thickness (see Figure 6); (2) the standard deviations of the displacements
(0)
(b)
20 -
? z loo-
;; ; l5-
“0 “0 to‘; 3 5-
“3 50-
I
#
2
6
I
I
1
10
14
16
h (cm)
Figure
6. Eigenfrequencies
h
versus the plate thickness.
(cm)
(a) Plate 1; (b) plate 2.
in selected cross-sections (see Figure 7 (h = 5 cm, p = 10(1/s)); curves la correspond to the first plate subjected to the single force, curves lb to the first plate subjected to the set of four forces, curves 2a correspond to the second plate subjected to the single force and curves 2b to the same plate subjected to the set of four forces, respectively; (3) the standard deviation of the displacement at a selected point (x = - 135, y = - 135) (the point where the seat of operator is usually located) versus the thickness of the plate (see Figure 8(a)); (4) the standard deviation of the displacement at the same point versus the damping factor (see Figure S(b)).
VIBRATION
,250
-150
-50
y
OF PLATES
0
50
WITH
STRAIGHT-LINE
EDGES
445
150
y
(cm)
(cm)
(d) 8
-250
-150
-50
0
50
150 -250
y (cm)
y (cm)
Figure 7. The standard deviation x = 50 and (s) section x = 150.
of the displacement
in (a) section
x = -150,
(b) section x = -50, (c) section
(01
10
T98 $7 : s x b
6 5 4 3 2 1 2
4
6
8
10
12
p(l/s)
Figure 8. The standard deviation and (b) the damping factor.
I
Figure
9. Square
h (cm)
of the displacement
plate with a square
at the point P(-135,
I
hole having
-135)
versus (a) the plate thickness
Clamped
1
clamped
outer and free inner edges.
446
K. POLTORAK
Results like these can be used as an aid when choosing the appropriate shape of the real plate, its thickness or location of the load. They show the convenience of the developed procedure for comparative analysis. However the necessity for numerical evaluation of the surface integrals (26) and (27) is a considerable obstacle in applying optimization techniques. In order to check the validity of the calculations the same computer program has been used also to evaluate the frequency parameters A of a typical square plate with a square hole having the outer edge clamped and the inner one free (see Figure 9). Since values of A for such a plate are available in literature the present results have been compared with those obtained by Nagaya [8] (see Table 1). The comparison shows satisfactory agreement of the results. TABLE 1 Comparison
offrequencyparameters
plate with a square
hole having
and free inner edges;
A of a square clamped
outer
A’/ A = 0.4
Mode
Reference [ 81
present results
1 2
3.456 3.905
3.460 3.918
5. CONCLUSIONS
A method for dealing with vibration problems of plates having both clamped and free, straight-line, mutually perpendicular edges and subjected to loads consisting of a set of transverse, arbitrarily located, random forces has been described in this paper. The expressions derived have enabled the author to prepare a universal computer program for any plate shaped and loaded within the imposed restrictions. The method seems to be useful in a great variety of technical problems since these restrictions are not very strong.
REFERENCES 1. H. B. KHURASIA and S. RAWTANI 1978 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers Series E 42, 215. Vibration analysis of circular plates with eccentric holes. 2. H. D. CONWAY 1960 Journal of Applied Mechanics, Transaction of the American Society of Mechanical Engineers Series E 28, 288-291. The bending and flexural vibrations of simply supported polygonal plates by point-matching. 3. H. D. CONWAY and K. A. FARNHAM 1965 International Journal of Mechanical Sciences 7, 811-816. The free-flexural vibrations of triangular, rhombic and parallelogram plates and some analogies. 4. P. A. A. LAURA 1968 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers Series E 35, 198. Discussion to the eigenvalue problem for twodimensional regions with irregular boundaries. 5. J. C. M. Yu 1971 Journal of the Acoustical Society of America 48, 781-785. Application of conformal mapping and variational method to the study of natural frequencies of polygonal plates. and G. SVED 1979 Journal of Sound and Vibration 67, 253-262. 6. D. Bucco, J. MAZUMDAR Vibration analysis of plates of arbitrary shape-a new approach. 1979 Journal of Applied Mechanics, Transaction of the American Society of 7. K. NAGAYA Mechanical Engineers Series E 45, 629-635. Dynamics of viscoelastic plates with curved boundaries of arbitrary shape.
VIBRATION OF PLATES WITH STRAIGHT-LINE EDGES
441
8. K. NAGAYA 1981 Journal of Sound and Vibration 14, 543-564. Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape. 9. V. L. RVATCHEV 1973 l’VreMethod of R-functions in Bending and Vibration Problems of Plates with Complex Boundaries. Kiev: Naukova Dumka (in Russian). 10. V. L. RVATCHEV 1982 Theory of R-functions and Some its Applications. Kiev: Naukova Dumka (in Russian). 11. A. WABERSKI 1978 American Institute of Aeronautics and Astronautics Journal 16, 788-794. Vibration statistics of thin plates with complex form. 12. A. WABERSKI 1980 Journal of Sound and Vibration 70, 453-457. A note on the analysis of non-stationary random vibrations of plates of complex form.
ofSound
and Vibration (1986) W(3),
VIBRATION
437-447
OF PLATES WITH
CLAMPED
AND K.
STRAIGHT-LINE,
FREE
EDGES
POLTORAK
Institute of Machine Design Fundamentals, Warsaw Technical University, 02-524 Warszawa, ul. Narbutta 84, Poland (Received
1 August 1984, and in revised form 16 February 1985)
This paper deals with vibration problems of thin plates having straight-line, mutually perpendicular, clamped and free edges and subjected to a load consisting of a set of transverse, arbitrarily located random forces. It is assumed that the number of edges of a plate forming recurring figures is optional but each of these edges is either clamped or free along its entire length. The procedure for solving the boundary problem based upon the R-functions method and for estimation of transverse displacements based upon the correlation analysis is presented. Numerical calculations are carried out for two example plates. 1.
INTRODUCTION
Many authors have investigated vibration problems of plates with irregular boundaries. So far however only relatively simple shapes of plates have been considered because the boundary value problems of such plates with mixed boundary conditions turn out to be very complex. For instance solid plates with arbitrary outer edges and plates with doubly connected arbitrary shapes have been investigated. For these problems general approximate methods such as finite element, finite difference and point matching methods [l-3] as well as conformal mapping techniques [4,5], the constant deflection lines procedure [6] or the Fourier expansion collocation method [7,8] have been usually applied. Recently a new so-called “R-functions” method has been developed by Rvatchev [9, lo]. By means of this method any shape of plate with homogeneous or mixed boundary conditions can be treated. However in case of very complex plates it yields very complex analytical expressions which differ for each shape of the plate. So, in general, a universal computer program for arbitrarily shaped plates cannot be established. Rvatchev’s approach is based upon the Galerkin-Ritz method. However the basis functions which in linear combination approximate the solution are determined in a specific way, so that they can satisfy complex boundary conditions of arbitrarily shaped plates. In the first step of this procedure a transformation of logical expressions describing the area of the plate into analytical ones by using properties of R-functions is performed. As a result one obtains “contour functions” w(x, y) satisfying the following conditions:
[WIG=o,
w2 0 within the plate area,
w E ck+‘,
awlan
= 1.
(la-d)
Here G means the edge of the plate and n the normal direction to the edge. Afterwards the contour functions are substituted into general formulae for basis functions corresponding to a given combination of boundary conditions. The form of the contour functions significantly influences the complexity of this substitution and the following Galerkin-Ritz procedure since they are subject in this process to troublesome multiple differentiation, which cannot be performed numerically. For example, in the case of a plate with partly clamped and partly free edges the required order of differentiation is equal to 5. 437 0022-460X/86/030437+
15 $03.00/O
0
1986 Academic Press Inc. (London) Ltd.
438
K. POLTORAK
It is the purpose of this paper to show that when some restrictions on the shape of the plate are imposed then simple universal recurrence formulae for the contour functions, in a suitable form for computational analysis, can be derived. Since they are universal, the multiple differentiation has to be carried out analytically only once. So any change of the shape of the plate, within imposed assumptions, will not demand changes in the once derived expressions. Investigations have been carried out for plates having both clamped and free edges. The restricting assumptions are as follows. The plates have only straightline, mutually perpendicular edges. The number of edges forming recurring figures is optional but each of these edges is either clamped or free along its entire length. In the second part of the paper the probabilistic response of such arbitrarily shaped plates is considered. Previously the R-function method was applied in probabilistic problems by Waberski [ 11, 121 who investigated the response of plates subjected to random loading having the character of a measurable time-space field. In this paper a load consisting of a set of concentrated, transverse, arbitrarily located random forces is taken into consideration. The number of forces is unlimited. The response of the plate is found by reducing the problem to the problem of a system of many inputs and one output, and on the basis of correlation analysis. A plate of example shape subjected to an example load is shown in Figure 1. AY
Figure
1. The example
plate subjected
to the example
load.
The procedure presented in this paper can be useful in a great variety of technical problems because the restriction on the shape of the plate boundaries does not seem to be very strong if one notices that techniques applied to shape elements often produce similar restrictions. Numerical calculations have been carried out for two example plates, being simplified models of excavator platforms subjected to loads coming from engines.
2. SOLUTION
OF BOUNDARY
PROBLEM
The basis functions simultaneously satisfying boundary conditions on both clamped and free edges, regardless of the shape of the plate, have the form [9] Hij=wl
I
&w:A,- 6(wiW:'wi)[D~(w:h)+6[w:(D,Ai+A,D,w,)lA,D,(w:)I 1
+(2- Y)[~,~~(w:~~)+~w:A~~~~w~II
(2)
VIBRATION
OF
PLATES
WITH
STRAIGHT-LINE
439
EDGES
where
dg = Pi(~lX)Rj(V,y),
A,=[~/~(~:+~:)I[~,(w:~~)+~~(w:~~)I,
Pi and Rj being Tchebyshev’s polynomials of the first kind, n1 and n2 normalizing factors in accordance with plate dimensions, v2 Poisson’s ratio, and D,, D2, Dj, T, and TZ differential operators defined by 1=k
,=k
T,J=
Dkf= C
c ,=(,
i=O
iiyi(g)‘(E)k-‘. (-‘)c:dxk:k{ (3)
w,(x, y) and w,(x, y) are contour functions. The function w, is related to clamped edges and the function w2 to free edges of the plate. In order to find universal formulae for the functions w, and w2 one first considers two types of infinite areas-a vertical and a horizontal belt-in the plane co-ordinate system (see Figure 2). The areas of the belts including their boundaries can be described by simple inequalities: for the vertical belt,
R,=-ab+(a+b)x-x*20
for the horizontal belt.
LA,= -cd + (c + d)y - y2 2 0
Figure 2. The vertical
and horizontal
(4)
belts.
With i the belt number, one can define Xi = -Uibi + (ai + bi)X - X2,
K=-c,d,+(c,+dJy-y2.
(5)
Then by use of logical functions one can create an expression for a rectangular area: fl,=f2,nR,,
or
R,, = Xi 2 0 n Y, 2 0
(I means “rectangle”,
i = 1, 2 . . . n). (67)
This area will be called the elementary area. By use of Boole’s function, defined as
s,(s)={; p:;},
(8)
expression (7) can be rewritten as fl,i=[S>(Xi)=l]n[S,(
Yj)=l].
(9)
In order to transform the logical expression (9) into an analytical one one introduces the R-functions, which have the property that their signs depend only on the signs of the
440
K. POLTORAK
arguments. In this paper the following R-functions have been used: XA y=x+
Y-(x2+
Y2)1’2,
xv
y=x+
Y+(x*+
Y2)“2,
x=x.
(10)
These correspond to the logical product, sum and complement respectively. Thus, instead of (9) one can write 0,i = Sz(Xi) A S2( Y) 2 0
(II)
and, upon utilizing the properties of Boole’s functions (8), 0,i = S,[ Xi A Y] 2 0.
(12)
R,i = Xi A Y 2 0,
(13)
This yields finally
or, in the case of the complement of the rectangle including its boundaries, OCr,i= Xi A yi 2 0
(CTmeans “complement of the rectangle”).
(14)
Now it is evident that the function wi=XiA
Yi=Xi+
Yi-(X:+
Y?)“‘=O
(15)
describes the boundary of the rectangle. Naturally wi=Xi A Yi=Xi
A Yi=O.
(16)
Now one can consider the area as a product of complements of n rectangles (Figure 3). If one denotes (xY)i = xi + r, - (Xf-t Y:)“*, (XY),i = (Xf+ Yf)1’2- Xi - Y
(c means “complement”),
Figure 3. The product of complements of n rectangles.
Figure 4. The product of interiors of m rectangles.
(17)
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441
EDGES
then in accordance with expressions (6)-( 14) one has, for this area, n pE,n= (. . -((G,l n J&J n f&d n - - -) A fL =((...(((XY),,~On(XY),2~O)n(XY),,~O)n’..)n(XY),,sO)
**(((XVc,, A(Jmc,,) A(X%3 A*. *IAwn,,1
= a(*
(18)
= (. . .(((xy)c,, A(xy),,) A(xy),,) A. *‘1A(xy)c,n 2 0, (pc means “product of complements”). replaced by the recurrence formula =
(XV,,”
Cxy)pc,ili=n
=
The last member of expression
txy)p,i
-I+
(xy)c,l
-[(xy)‘,c,i-l
+
(18) can be
(xy)zC,i11’2~
i=l,2 ,..., n, (19) where (XY)PE.I = (XY),,. A similar procedure can be repeated for the area formed by the interiors of m rectangles with their boundaries (see Figure 4). As a result one obtains (XY),,
= (XY)p,ili=, =(XY),i-l+(XY)i+[(XY)i,i-l+(XY):]”’,
(20)
where (XY),, = (XY), (p means “product”). One can then form one more area, being a product of L$c,n and LZ,,,. Then the boundary of this area is described by the equation (XY) = (XY),,,
- [(XY)&, + (xv);,,]“’
+ (XY),,
= 0.
(21)
By making use of the function (XY) one can obtain the functions wi and w2 for any plate shaped in accordance with the general assumptions presented in the introduction to this paper. This is due to the fact that the function (XY) satisfies conditions ( la, 1b, 1c). Condition (Id) can also be fulfilled after simple modification of (XY). It can be proved that condition (lc) will be satisfied by w1 or w2 if the same condition is satisfied by all functions (XY)i: i.e., the functions describing boundaries of elementary areas used in the procedure. The modification can be easily carried out by rewriting functions (5) in the following way: Xi=[l/(bi-a,)][-aibi+(ai+bi)x-x2],
Y=[l/(ci-di)][-c;di+(ci+di)y-y*]. (22)
Having the functions Hij that satisfy boundary conditions one supposes that the solution of the equation of motion is of the form
U(-%Y, t) = f
i+j=O
Hc(X,Y) Kj(t)v
(23)
or, upon replacing the index matrix by the index vector for simplification of expressions,
UC% Yv t) = f &(X3 Y) V(t),
n=(m+1)2.
(24)
i=l
The equation of motion is transformed reference [ 111, into the set of equations
i,
{
V,D[Hi, 41 +
by means of Galerkin’s method, according to
(2pbP$+ Pb!$)}(H. Hj) = 11 .f(K
Y,
t)Hj dx dY,
R
j=1,2
3. . . n,
(25)
442
K. POLTORAK
where h is the thickness, p the mass density, p the damping factor, f(x, y, t) the applied dynamic load, D = Eh3/ 12( 1 - Y’) the tlexural rigidity, E the Young’s modulus, 0 the plate area and d2Hia2Hj
[H,, Hjl =
a2Hid2Hj
sa42+i)~2~-2
a2Hi d2Hj ax dy ax ay
n
dx dy,
HiHj dx dy,
(Hi, Hj) =
(26)
(27)
R A being the Laplacian operator. Eigenvalues can be calculated from the determinant of the symmetric matrix:
det I[Hi, HI] - A‘( Hi, Hj)I = 0.
(28)
The natural frequencies are wj = A,(D/ph)“*/A*,
(29)
where A denotes herein a chosen characteristic in-plane dimension of the plate. Equations (25) can be solved separately one by one only in the case when functions Hi satisfy orthogonality conditions: (30)
The functions Hi that satisfy condition (30) can be denoted by Gi. Actually they 1 are normalized eigenfunctions of the problem. Thus, one can obtain
& =i
j=1
avHj
I? ati%, i
j=l
where the av are coefficients to be determined equations
[LLjl{a,-I= 0,
j=l
a&
, >
from the set of homogeneous
i=l,2,...n,
(31) matrix (32)
where LLj = [ Hk, Hj] - A:( Hk, Hj). Each of these equations can be solved with accuracy to within a multiplicative constant by transformation into a non-homogeneous set of n - 1 equations, by putting air = 1 and applying Cramer’s formulae. 3. ESTIMATION OF TRANSVERSE DISPLACEMENTS By estimation of transverse displacements is meant herein the determination of displacement variances for selected points of the plate. The load, consisting of a set of transverse, arbitrarily located, random and mutually correlated forces, can be represented as
f(x, Y, t) = I, 5(t) 6(x, --xl S(Yr -Y),
(33)
where Z,(t) is a random force, s is the number of forces, S( ) is Dirac’s delta function, and x, and y, are the co-ordinates of the force application points. Then, by making use of the Dirac delta function properties the right-hand side of equation (25) can be
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in the following way:
6(&-X)
S(Yr-Y)Hj(X,
Y)
dx
R
dY=ir<1Zr(t)Hj(Xn Yr).
(34)
Thus, the set of equations (25), upon taking into consideration e_quation (34) and replacing the functions H(x, y) by the corresponding eigenfunctions H(x, y) (see equation (31)) can be given in the form (35) Of course, now each of these equations can be solved one by one because terms other then those in which i =j, according to the orthogonality of the eigenfunctions, have vanished. The steady state solution is given in the form cc
I
G,( T)Z,( t - T) dr,
(36)
0
where Gj = (l/p,) e-+7 sin Pjr is the Green function, pj = (~3 - P~)“~ for w > )(L,and T is the time difference. Finally the displacement function can be expressed by Gj( T)Zr(
and the correlation
t
-
T) dr.
(37)
function for a chosen point P(x, y) of the plate by
K,(x, Y, 7) = & X
i$, ,il r$l “$, tTl,Cx9 Y)fiCx9
Ylfiitxn
Gi(T1)Gj(T2)RN(T-T2+T,)
Yr)fiCxw
dT1
dT2,
Yu)
(38)
where r,u=l,2 ,..., s, and R,( T - TV+ TV) are the mutual correlation functions of the random forces. The variance of the displacement can be easily obtained from formula (38) by putting T = 0. 4. NUMERICAL EXAMPLES In order to illustrate the use of the proposed procedure numerical calculations have been carried out for two example plates subjected to two variants of load. The first plate (see Figure S(a)) is rectangular and has an inner rectangular hole. The second one (see Figure 5(b)) is also rectangular but has nine inner rectangular holes. Both plates are clamped on the edge of the central hole. The remaining edges are free. These plates are simplified models of excavator platforms. In the first variant the plates are subjected to a single random force and in the second variant to a set of four forces. The dimensions of the plates and the locations of the forces are shown in Figure 5. The material data, necessary for some calculations, have been chosen as for steel. The mutual correlation
444
K. POLTORAK Y
t
-150
-50
50
150
x
-75
?? A ?? Gl
‘I, i,
i, -250 (a)
(b)
Figure 5. Two example plates investigated. of four forces. (a) Plate 1; (b) plate 2.
Dimensions
in cm. Z,, the single force;
Z,, Z,, Z,, Z,, the set
functions of the applied forces are assumed as R, = (1/2)“(u,+a,)
eC_T-p~a~,
‘=
a, and a, being the standard deviations of the random forces, LX,.,, and &-coefficients determining the degree of mutual dependence, and a,.,, the distances between the forces. These functions have been used to determine the variance of the displacement by substituting them into equation (38), executing the integration and putting r = 0. In these examples u,, a,, u3, a, = lo3 N, all coefficients (Y,,, = 15 x 103(l/s), and all- coefficients (Y, = 0.7. The following results have been obtained: (1) the relation between eigenfrequencies and the plate thickness (see Figure 6); (2) the standard deviations of the displacements
(0)
(b)
20 -
? z loo-
;; ; l5-
“0 “0 to‘; 3 5-
“3 50-
I
#
2
6
I
I
1
10
14
16
h (cm)
Figure
6. Eigenfrequencies
h
versus the plate thickness.
(cm)
(a) Plate 1; (b) plate 2.
in selected cross-sections (see Figure 7 (h = 5 cm, p = 10(1/s)); curves la correspond to the first plate subjected to the single force, curves lb to the first plate subjected to the set of four forces, curves 2a correspond to the second plate subjected to the single force and curves 2b to the same plate subjected to the set of four forces, respectively; (3) the standard deviation of the displacement at a selected point (x = - 135, y = - 135) (the point where the seat of operator is usually located) versus the thickness of the plate (see Figure 8(a)); (4) the standard deviation of the displacement at the same point versus the damping factor (see Figure S(b)).
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,250
-150
-50
y
OF PLATES
0
50
WITH
STRAIGHT-LINE
EDGES
445
150
y
(cm)
(cm)
(d) 8
-250
-150
-50
0
50
150 -250
y (cm)
y (cm)
Figure 7. The standard deviation x = 50 and (s) section x = 150.
of the displacement
in (a) section
x = -150,
(b) section x = -50, (c) section
(01
10
T98 $7 : s x b
6 5 4 3 2 1 2
4
6
8
10
12
p(l/s)
Figure 8. The standard deviation and (b) the damping factor.
I
Figure
9. Square
h (cm)
of the displacement
plate with a square
at the point P(-135,
I
hole having
-135)
versus (a) the plate thickness
Clamped
1
clamped
outer and free inner edges.
446
K. POLTORAK
Results like these can be used as an aid when choosing the appropriate shape of the real plate, its thickness or location of the load. They show the convenience of the developed procedure for comparative analysis. However the necessity for numerical evaluation of the surface integrals (26) and (27) is a considerable obstacle in applying optimization techniques. In order to check the validity of the calculations the same computer program has been used also to evaluate the frequency parameters A of a typical square plate with a square hole having the outer edge clamped and the inner one free (see Figure 9). Since values of A for such a plate are available in literature the present results have been compared with those obtained by Nagaya [8] (see Table 1). The comparison shows satisfactory agreement of the results. TABLE 1 Comparison
offrequencyparameters
plate with a square
hole having
and free inner edges;
A of a square clamped
outer
A’/ A = 0.4
Mode
Reference [ 81
present results
1 2
3.456 3.905
3.460 3.918
5. CONCLUSIONS
A method for dealing with vibration problems of plates having both clamped and free, straight-line, mutually perpendicular edges and subjected to loads consisting of a set of transverse, arbitrarily located, random forces has been described in this paper. The expressions derived have enabled the author to prepare a universal computer program for any plate shaped and loaded within the imposed restrictions. The method seems to be useful in a great variety of technical problems since these restrictions are not very strong.
REFERENCES 1. H. B. KHURASIA and S. RAWTANI 1978 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers Series E 42, 215. Vibration analysis of circular plates with eccentric holes. 2. H. D. CONWAY 1960 Journal of Applied Mechanics, Transaction of the American Society of Mechanical Engineers Series E 28, 288-291. The bending and flexural vibrations of simply supported polygonal plates by point-matching. 3. H. D. CONWAY and K. A. FARNHAM 1965 International Journal of Mechanical Sciences 7, 811-816. The free-flexural vibrations of triangular, rhombic and parallelogram plates and some analogies. 4. P. A. A. LAURA 1968 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers Series E 35, 198. Discussion to the eigenvalue problem for twodimensional regions with irregular boundaries. 5. J. C. M. Yu 1971 Journal of the Acoustical Society of America 48, 781-785. Application of conformal mapping and variational method to the study of natural frequencies of polygonal plates. and G. SVED 1979 Journal of Sound and Vibration 67, 253-262. 6. D. Bucco, J. MAZUMDAR Vibration analysis of plates of arbitrary shape-a new approach. 1979 Journal of Applied Mechanics, Transaction of the American Society of 7. K. NAGAYA Mechanical Engineers Series E 45, 629-635. Dynamics of viscoelastic plates with curved boundaries of arbitrary shape.
VIBRATION OF PLATES WITH STRAIGHT-LINE EDGES
441
8. K. NAGAYA 1981 Journal of Sound and Vibration 14, 543-564. Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape. 9. V. L. RVATCHEV 1973 l’VreMethod of R-functions in Bending and Vibration Problems of Plates with Complex Boundaries. Kiev: Naukova Dumka (in Russian). 10. V. L. RVATCHEV 1982 Theory of R-functions and Some its Applications. Kiev: Naukova Dumka (in Russian). 11. A. WABERSKI 1978 American Institute of Aeronautics and Astronautics Journal 16, 788-794. Vibration statistics of thin plates with complex form. 12. A. WABERSKI 1980 Journal of Sound and Vibration 70, 453-457. A note on the analysis of non-stationary random vibrations of plates of complex form.