RECTANGULAR ELASTIC THIN PLATES. Fu Bao-han ( N~i! ) (Northeast Heawy Machinery Institute. Qiqihaer). (Received May. 5 1934 Communicated by Chien ...
Applied Mathematics and Mechanics (English Edition, Vo1.6, No.ll, Nov. 1985)
e
Published by ~ U 1 , Shanghai, China
.
APPLICATION OF THE RECIPROCAL THEOREM FOR CALCULATING THE NATURAL FREQUENCIES OF RECTANGULAR ELASTIC THIN PLATES Fu Bao-han ( N~i!
)
(Northeast Heawy Machinery Institute. Qiqihaer) (Received May. 5 1934
Communicated by Chien Wei-zan g)
Abstract This paper further extends the applications oj the reciprocal theorem 10 calculating the natural frequencies of rectangular elastic thin plates on the basis of [If. Applying the presented method. there is no need to so/we gowerning differential equations. it is only necessary to so/we a simple integral equation after using the reciprocal theorem between the basic syst~m. and the actual system. Using the idea of the generalized edge simply supported and introducing the frequency matrix. tlten all frequency equations of the rectangrJiar plates with two opposite edges simply supported and other two opposite edges variously supported are obtained together. This is a simple. convenient and general method for calculating the frequency equations of the rectangular plates.
1. introduction The reciprocal theorem is a classical theorem, i,[2J points out that between two identical linear elastic bodies, if states of their stress and strain are actual, then the reciprocal theorem is yielded whether their boundary conditions are the same or not. However, as far as [I] know, s~ far [3] the reciprocal theorem has not beCn applied between any two bodies with e\'~dently different boundary conditions and any kind of problems has not been systematically sovled by means of it. [3] has first applied the reciprocal theorem between two rectanrular plates with evidently different boundary conditions and has solved the deflection equations of rectan~ular plates with various edge conditions. [3] has presented a systematical methvd for solving the bending rectangular plates. [4] !,las pointed out that the reciprocal theorem is equiviU(!nt to the superposition principle of displac,ements under certain condition, then why obtai~ solutions by means of application of the reciprocal theorem to solving the- ~~nef~tj0n equations of the recU!ngular .plates which are indenclcal with those hv means of the superposition principle of displacements on the basi:; of analytic h,et~od had been illustrated. This new discovered function of the reciprocal theorem had provided an effective method for sOlving deflectiori equations of the rectangular plates with various edge conditions!··5f. [1] had extended the idea of [3, 4] to calculating the natural frequencies of the sl~aight ~ams and this paper appli~ the method of [IJ to calculate the natural frequencies of the rectangular plates. For this method, we needn't to solve the governing differential equations but we need only to
1069
Fu Bao-han
1070
solve a simple integral equation after using the reciprocal theorem between basic system and actual system. Using the generalized edge simply supported and introducing the frequency matrix, all .frequency equations of the rectangular plates with two opposite edges simply supported and other two opposite edges variously supported are obtained together.
II. Elementary Theorem From the Kirghoff assumptions we know for the elastic thin plates there is the governing differehtial equatiOll[6]
q(x,Y)
D
(2.1)
where q(x, y) is the applied static loading. In the free vibration of a plate there will be no surface loading q(x, y) and q(x, y) must be
a replaced by -PafW(x ,y ,t) 2
from the d'Alambert principle, we thus obtain the governing
differential equation in the form
a'W(x,y,t) ax'
a -n1 ~W(x,y,t) 1
+2 8'W(x,y,t) + a'W(x,y,t) ax1ay2
8y4
(2.2)
assuming W(x, y, t) =w(x, y) T(t) and substituting it for (2.2), we obtain 4
~r. 8'w(x,y,) +2 a w(x,y) + 8'~(7'Y) ]/W(X,y) 2 p
Lax'
8x ayl
y
(2.3) It can be firmly believed that from the left-hand side of eq. (2.3), after some rearrangments we obtain a'w(x,y) ax'
+2 a'w(x,y) + a4w(x,y) Z ax ay2
ay"
~pc.>1w(x,y)
u
(2.4)
where w(x, y) is the solution of made shape eq. (2.4) and the quantity c.> is circular frequency. Comparing the right-hand sides of (2.1) and (2.4), and applying the reciprocal theorem between the is looked on as . unit load basic system and the actual mode shape systems when pc.> 2w( x, y ) external loading, we shall finally find the solutir)O of the mode shape equation and frequency equations.
DI. The Frequency Matrix of the Rectangular Plates with Two Opposite Edges Simply Supported Calculating the natural frequencies cl the rectangular plates we shall find the solution of the mode shape equation in free vibration. In the following let us apply the reciprocal theorem to find the- solqtion of the mode shape equation of the rectangular plates with two opposite edges simply supported and other two opposite edges variously supported. A simply supported rectangular plate as shown in Fig. I is acted by a unit of concentrated load at floWing coordinate point (g, TJ) called the unit load basic system or basic system. The rectansular plate with two opposite edges simply supported and other two opposite edges generalized slmply supported, as shown in Fig. 2, considered as acted by pc.> 1w( x, Y ) is called
The Natural Frequencies of Rectangular Elastic Thin Plates
1071
~ctual
system of the mode shape or actual system. w,o{x), M,o{x), w,.{x) ,M,.{x) \lenotes the deflections and the bending moments along the two opposite generalized edges simply ~pported y=O and y=b respectively. . Applying the reciprocal theorem between the two systems as shown in Figs. 1 and 2, we obtain ~,f])dxdy
(3.1)
('l
I;-=-=-~-=-==---
1'7
II
I I
-
r
I
I
/;
Wyo(X)
x(E)
0
I
I I
b
II
r
I
I I
M)'b(x)
~
xCO
Mye(%)
I
wn(x) 1I(,,)
11(10
Fig. 1
Fig. 2
From [4J we have
X l I . mtrx
.~,2 . nrry
.
mtre
• SID-- S l n - -
b
SiD mtrx
a
.
.
a
mrr~
.
.---'=+~~""":-_)-"1 Sln---iz--
-(-:-;:' __
.
Mf]
(3.2)
SI~
b'
nrrrz
SlD---SID--
mrrx
a
.
mtr~
. cosnrr SlD-~- Sln-a
(3.3)
b
.
nrrT]
(3.4)
Sln---z;-
and assume DO
Wa,TJ)=
w{x,y)=
E
oundary conditions, we obtain respectively
(3.12) (3.13)
1074
Fu Bao-lian
rmr
+b.{[ a!-(2-v) ( -a-
)1] ( a",cha.b;
-
sha.. -
2
t
a. Sha.;) b cha.. -2
mlr )2] f3 .. chf3... ~ - P• -(2-v)-a b shf3 ..y %
(
[
b
1
+ D A.. (
- iJB· 1
a.cha_:2
b
sha.. -
b
b
+
a.sha .. :f
P. chf3..?;
cha.-
shP"2"
b
2
b
2
b a.sha.2"
(a.Cha.;
b
b cha~
sha.-
shf3.-
chf3.-
2
2
2
P.:)_
f3 ..chf3. : f3. Sh '--""""""b.-=- + b
-
0
(3.14)
2
t
( mn + [P2- ( 2 - v ) -)1 ] f3.shf3. ; • a b chf3.2
+b..
2
{[
(
mlr
a .. -(2-v) -a-
_
)2] ( a",chamb; sha m -
a",sha m ; )
+----b"-eha",-
2
2
mlr )2] (f3 .. f3 ..b: + f3. ShP..b;)} - [ f3!-(2-v) (~ Ch
shf3 .. -
ehf3..-
2
+
1
D A ..
b a",sha"2"
~
(a.eha ...
b
b
sha ..-
-if3'"
(a",eha.,
~
b
sha.2"
b
f3", chf3.. Z
b
eha ..-
2
1
2
shP.2
2
+
b a",sha"2"
b
eha..-
2
P",ehP",f b shP.--
b chf3.2
P.Sh/3.;).
b
chf3.-
2
~2]Z a.. ( chaM b;
z ( mn -a.. {[ a.-(2-v) - a
+
' b)
f3 .. sh f3"'2"
sha .. -
2
2
sha.. ; )
+--~-
b
cha.--
2-
=0
(3.15)
-
The Natural Frequencies of Rectangular Elastic Thin Plates
2 11IJT j J2 ch 13.: )}. 13". (Sh 13.;) b +-~b~ [ f3 ..1-(2-v) (-a-7
-
chf3".-
shf3.. -
2
2
mrr )1 J1 (Ch a ...b~ a.. {[a ..2-(2-v) (-a-
+b".
sh a. 2b ) ----rb-
cha".--
sha".2
mJr )2 J2 13m (Ch 13mb; [ f3 1.. -(2-v) (-a-
-
2
sh fJ ...b; )}
-
shf3m-
chf3..-
2
1
mrr
2
2
cha"'2 b b ( sha".-
2
+ DAm [ar.-(2-v)(-a-) Jam {
f.. mJr )1] 13". ( [f3 ..2-(2-v\-a-
-
mJr J D B "'la!-(2-V)(-a)
~
sh 13m
1
2
)2J 13m (
mn f3 ..2 -(2-v) ( -a-
{[ 2
(
Jam
mJr
2
b
(
sha".-
cha.-
2
chf3.b
2
~
shf3,.,b
shf3.-
2
sh
-
sha",-
f3 . : .+ [.2 P.. -(2-v) (mtr)1Jl -a13. (Sh' b
J
(
mn
chf3.2
2 ( mn f3.-(2-v) -a-
shf3·
sha~
+
b
2
sh a .. ; )
b cha.2
shf3.
~
)}
+-----rb~
chp.-
shP~
2
1 A {[ ) J] i:c. (• a!-(2-v) ( (mn +n J
)
Ch f3. ; ) }
-
)2 J2 a.. (Ch a ..b~
)2]1 13. (Ch f3..b~
a.~b 2
2
[
)} =0
cha..-
2
+b ..{ [ a.-(2-v) - a
~
chf3",-
)1 J1 a. (Ch a",b. :
,
)}
sha-f)
b cha ... 2
2
-a .. a,.-(2-v) -a-
-
b cha".-
ch 13.; b shf3" 2
+
b \ chf3.2
[
sha"'2 b )
+
2
-
-
1075
2
ch a.b;
sha.2
(3.16)
-
Fu Bao-lian
1076
~
1 {[ 2 ( mlf \' ] (Ch a ..b - DBm a .. -(2-v) - aJ am
sha..-2
( mlf
)2J P.. (
- [ P!-(2- V )'-1j
ch,8...
~
b
+
~~
sh a ..b cha..-2
+
sh,8..
~
b
)}
(3.17) =0 chP...-.:2 If [i,11 (i,j= 1, 2,3,4) denotes the corresponding coefficients of am. bm. Am and Bm, (3.14)-
shp..2
(3.17) may be written in the simple forms [1, 1Ja",+[1,2]b .. + [1 ,3JA",+ [1, 4JB", = 0
(3.14)'
[3, 1]a.+[3 ,2Jb",+ [3,3JA.+[3, 4JB.=0
(3.15)' (3.16)'
(2 . 1 Ja ..+ [2 ,2]b.. + [2, 3JA .. + [2 ,4]B",=0
[4,lJa.+ [4, 2]b", + [4, 3JA.+ [4, 4JB.=0 Arranging [i. 11 in the matrix and denoting it by the sign
~= f
l
We call
!l
(3.17)'.
Ll ,we arrive at
[1,1]
[l,2J
[1,3J
[2,lJ
[2,2J
[2,3J
[1,4J ] [2,4J
[3,lJ
[3,2J
[3,3]
[3,4J
[4,1]
[4,2J
[4,3J
[4,4J
(3.18)
the frequency ma.trix.
IV. The Applications of the Frequency Matrix to Solving the Frequency EquatiolUl Using the general~ edges simply supported and then introducicg the frequency matrix are very convenient for solving frequency equations of the rectangular plates with two opposite edges simply supported and other two opposite edges variously supported. The zero determinants formed from different elenttnts of the frequency matrix indicate frequency equations with corresponding boundary conditions. There are asymmetric, symmetric and antisymmetric modes of axis which pass through the center of the plate and are parallel to x axis. Let us discuss the three cases in the following respectively. (1) Asymmetric modes 1. 11 = 0 simply supported and fl. = b clamped. For this case, the boundary condition (3.15) must be enforced. Setting a .. =b.=A .. =O and B.~O and taking the element [2, 4] of I:l equal to zero, we obtain the frequency equation
a .. cha.bsinp~b-,8~sha ..bcosp~b=O where
(4.1)
-
The Natural Frequencies of Rectangular Elastic Thin Plates
j
IF>( man =
2
and
fJ~2=}.2
_( ~rr )
2
=
2. 1] 0 simply supported and T/ b free In this case, the boundary condition (3.17) must be satisfied. Setting
b.:IIp 0 and taking the element [4, 2] of 6.
and
1077
equal to zero, for
r.
a",=A",=H .. =O
IF < ( "~~
the
frequency equation becomes
for
>(
A.2
:n ) 2
,one becomes
+fJ~[fJ~2 +(2- V )(
= 0 clamped and
:Jr YJ[a!-v( n;; r]shaMbcosP~b=O
(4.3)
= b free It is necessary to satisfy the boundary conditions (3.14) aand (3.17) together. Settin, a.-B.. =O and b.~O, A.~O and using zero determinant of the elements [I .2It [1.3); [4.2], 3.
f1
[•. 3] of
l:!. • for
A.~«
7}
m; ) l
•
the frequency equation ~omes
{[a!-v( marr YJ cha.,b-[fJ!-vC:n )'}hP.b }
:n
· {[a!-(2-11}(:lr r]cha b-[fJ!-(2- V)( YJchfJ..b } lr n -{[ a;-v( "'a YJ shaMb-(T)[fJ!-v("'a YJshfJmb } · {[a;-(2-v)("'an shamb-( ~: )[fJ~-(2-V)( "'alr fJshfJ. b }=O (4.4) ).2 >( "'aT( 12 ,one becomes M
rJ
for
{[ a;-v( m: YJ cha.b+[fJ~2+v(:rr r}osfJ~b}
. {[a;-(2-v)( ~rrr]cha..b+[fJ~1+ (2-v)~"'an r)osfJ~b} -{[a!-v("'arr fJshamb+(-~_)[fJ~2+v("'an rJsinP~b} · {[ a!- (Z-v)( ": YJ sha..b-(
!: )[fJ~2+ (2-v) ( "'an YJsinfJ~b}=O
(4.5)
-
Fu Bao-lian
1078
We must take n= 1,2, 3, ... for the three asymmetric cases.
(2) Symmetric modes 1. Two opposite edges clamped Enforcing the boundary condition (3.14) or (3.15) and setting A",=B",:\=O , we obtain from (3.18)
am = b.. = 0
{[I,3]+[I,4]}=O
(4.6)
and then frequency equation becomes (4.7)
2. Two opposite edges free Enforcing the boundary condition (3.16) or (3.17) and seting a",= b", ~O , we obtain from (3.18)
For .11
«
{[3,1]+[3,2J}=O
:1r )
~Jr ) 2
(4.8)
m; ),J[ P!-v( :1r )'] sham ~ cht3. ;
+ t3.[P!- (2-V)( :Jr )'][ a! - v ( :Jr ,12> (
and
,the frequency equation becomes
Z
-a..[a!-(2-v)(
for
A.:=B",=O
Y] ch a... ; sht3... ~ =0
(1.9)
one becomes
,
)2][!3~2+V (mJr b b -a- ')2] sha"zcos!3!T
mJr a. [ a!-(2-v) (-a-
For the two symmetric cases we must take n= 1,3,5, ...
(3) Antisymmetric modes
1. Two opposite edges clamped Enforcing the boundary condition (3.14) or (3.15) and setting A.= -B .. :\=O ,we obtain from (3.18) {[1,3]-[1,4]}=0
a",=b.=o
and
( j.ll)
and then the frequency equation becomes
b. b b , b_ a ..cha ..-sln.8!- - .8~sha.-2 cos.8. -2 - 0 22·
2. Two opposite edges free Enforcing the boundary condition (3.16) or (3.17) and setting a.= -b.. ,.O ,we obtain from (3.18) For A:
« '"air )
{[3,1]-[3,2]}-=O I
,
the frequency equation becomes
(4.12)
A.=B.. =O
and
(4.13)
The Natural Frequencies of Rectangular Elastic Thin Plates
( mJr )' -am [ a!-(2- v h;-a-
]-l. POI-V ' ('11IJr -a- )
, mrr ) + {3" [ {3~- (2-V)(~
for
IF
>( m: )
%
2
2
1079
b , .. 2 b Jcha"'zshp
21
Jl- a~-v ( m:n )" Jsham Zchf3",Z=O b b -0-
(4.14)
,one becomes
For the two antisymmetric cases we musi take n = 2, 4, 6, .. , For (1), (2) and (3), we must take m= 1,2,3, ... The previous frequency equations obtained by us are all identical with these of [7]. APPENDIX For calculating the natural frequencies of the straight beams and rectangular plates, we shall give some transforms from trigonometric series to the hyperbolic functions in the following. For the simply support beam on elastic foundation under combined action of lateral load and tension, as shown in Fig. 3, there is the diflerential equation
Fig. 3
(A,n
Denoting N 21]= E!'
the eq. (AI) becomes
when
T]1> pi
, the general solution of (A2) is w(x) = Asha..:+ Bchax+Cshfi,"+ IJchf3x
where
:,1\,3)
1080
Fu Bao-lian
The solution (AJ) must satisfy the boundary conditions
w(+)=o.
(A.4)
Enforcing the boundary conditions (A.4), (A.3) becomes
.l:¥.
w" _
'J,U
(h) - -
al -
1
Pi
(_ shax
+
chax
-:-fh s a-- --:-Th c a2
+
2
shflx _ chflx )" -;-::rhfl -;-::rhfl s _ c 2 2
(A.S)
Taking the coordinate axes as shown ht Fig. 4, (A.5) becomes
N
N"
x
Fig. 4
sha(x-f)
1
sha2
+
cha(
x-f) 1
cha2
+~(,,~ ~) - Chfl("~ ~) J shfl-
(A.II)
chfl-
2
2
The problem can be solved by means of the energy method. too. For the coordinate UC$ as abown in Fig. 4, the total potential energy is equal to
np-=1EJ~:( ~:. rdx+~(~~ rdx++~"wld"-M.(~~)._.
(Aj)
Aiiuming
we,,) =
E 00
(A.S)
and using the principle of minimum potential energy, we find (A.9)
Comparing (A.6) and (A.9), we obtain
~
"'~1,1
+
m'+
m~ msin--yI' 11 pL.-. n' n4
2T}ml--+
cha( x-~)
t
cha2
+
--1..-[_ 4F
__ n3
-
shfl( x--i) sh~ 2
sha( x- ~) 2
al_fll
chfl(
X-+)
chfl-I 2
h
I
2
s a-
(A.I0)
The Natural Frequencies of Rectangular Elastic Thin Plates When the straight beam is subjected to an external end moment Mo at x = manner, we obtain the transform in the form
cha(
/,
1081 using the similar
x--i)
( A.I\)
1 cha2
Forming the second derivatives of (A. 10) and (A.Il) to x, we obtain respectively
(A.12)
and
/J2chP( x chpl
+)]
(A.13)
2.
References [1] Fu Bao-Iian, A new method for calculating the natural frequencies of lateral vibration of the straight beams, Journal of the Northeast Heavy Machinery Institute, 3 (1983). (in Chinese) [2] Chien Wei-chang and Yeh Kai-yuan, Elastic Mechanics, Science Publishing House (October, 1956). (in Chinese) [3] Fu Bao-Iian, A new method for solving deflection equations, The Third Scientific Report CO[lferece of the Northeast Heavy Machinery Institute (january, 1981). (in Ch{nese) [4] Fu Bao-Iian, Application of the reciprocal theorem for solving equations of the deflection surface of the rectangular plates with various edge conditions, Applied Mathematics and Mechanics, 3, 3 (June, 1982). [5] Zhu Van-bin and Fu Bao-Iian, Further research on the bending of cantile.ver rectangular plates under a concentrated load, Journal of the Northeast Heavy Machinery Institute. I (1984). (in Chinese) (6] Timoshenko, S. and S. Woinowsky-Krieger, Theory of Plates and Shells. Second Edition (1956). [7] Gorman, D. J., Free Vibration Analysis of Rectangular Plates, Elsevier, New York, Amsterdam, Oxford.
