تحليل ارتعاشات آزاد ورقهاي بيضوي ساخته شده از مواد FGM
شاهرخ حسيني هاشمي1، سيد رسول آتشيپور2*، محمود كريمي3، مهدي اسحاقي4

استاد، دانشكده مهندسي مكانيك، دانشگاه علم و صنعت ايران
دانشجوي دكتري، دانشكده مهندسي مكانيك، دانشگاه علم و صنعت ايران
كارشناس ارشد ،شركت مهندسي و ساخت توربين مپنا (توگا)
دانشجوي دكتري، دانشكده مهندسي مكانيك، دانشگاه كنكورديا
چكيده
) %* ( FGM !”# $ %& ‘
: ; : 6:< : $ 2:= $ : >!:? + 9!%# 7+ 8 .6′ $ 4 ’23 1 0% / “, !- . + : F B E B 9! ’23 D !% ( 3 6′ 6 < $ 6’ 9 BC @A 3 #
9A 8 L A < K ‘ %’ + J FGM 9F A F H & 3 6′ $ I # . E < $ G%’ $ ‘ $ E :< 9 :’ $ S “= 0% / “, !- . 9 R” . !39 P 6 *Q %’ $ A ‘ 9″MN O3
R” 8 ! , V U N W %A 8 , O T MA $ 4 N 6 U 6′ %# E
T :MA N 6 U L ” W %A Q N U N W %A O FA “Q .6′ $ 9’ FGM
1268732112263

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

9 : BC @A 3 # H & 9!R” Y % + ‘ ! X L + . !39 FGM $ L : % /L : % + L : % + /L : % L : % /L : % ; :N Z F 6′ %# E < V .6′ $ 4 L % + /L % + فركانس طبيعي، روش چند جملهاي- ريتز ،FGM كلمات كليدي: ورق بيضوي، مواد هدفمند

Free Vibration Analysis of Functionally Graded Elliptical Plates

Sh. Hosseini Hashemi1, S.R. Atashipour2, M. Karimi3, M. Es’haghi4

Professor, School of Mechanical Eng., Iran University of Science and Technology
PhD Candidate, School of Mechanical Eng., Iran Univ. of Science and Technology
MSc in Mechanical Eng., MAPNA Turbine Eng. and Manufacturing Co. (TUGA)
PhD student, Department of Mechanical Engineering, Concordia University

Abstract
This paper deals with a free vibration analysis of functionally graded elliptical plates with different classical boundary conditions on the basis of polynomial-Ritz method and classical plate theory. The proposed admissible function is capable to obtain accurate natural frequencies of various classical boundary conditions namely, clamped, free and simply supported edges. The mechanical properties of the FG plate are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. The convergency of applied polynomial-Ritz method is investigated. In order to verify the accuracy of the present solution, a comprehensive comparison study is performed
[email protected] نويسنده مسئول مقاله *
with the available results in the literature for homogeneous elliptical plate and FG circular plate. Also, the numerical results of the present solution for FG elliptical plate are compared with those of the finite element method (FEM). Finally, the effects of the plate geometry and inhomogeneity on the natural frequencies are investigated and some vibrational mode shapes related to symmetric-symmetric, symmetric-antisymmetric, antisymmetric- symmetric and antisymmetric- antisymmetric states are presented.
Keywords: Elliptical plate, Functionally Graded Materials, Natural Frequency, polynomial-Ritz
method

7E [`] ` qA 9 . %’; %O %O FGM o A
GU r & GU f! 6
[u] qA [t] !s” . A3 9′
1O ;2%& . o A 1 $ G%’
FGM fA “3 O+ v 0 9 M ,
$ > . A”A 9 N 6 9 ” A . E F [w] tfA F” o A L % A 9V& Y
3 9 V . A < V FGM 9″ 9! ON 6′ %# 7+ U 9E o A fA “3 ufA F” %’; %O FGM D V< N
(wDQM) Q G < N . $ G%’
.[^ _] A$ < 9′
S F 9 9N C O
L+ 9 =2C L 6′ H & 9′ !
$ L , L 9′ ‘ @A 3 #
9 , =2C FA , .6′ 9 , 9 V%O
%# E < , %”3
< 9% !U 3 D A .6’
+ GU L9 “, L 3 !%O 9>,
– f3o OF GU 9A x <
8 !U $j 9N S B $ %# E F
=2C [[[ [y] _O .”A $ ‘9%?3
( L + 93
1268732112263

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

9′ ‘ @A 3 # “A 4 $ ‘ E L @+ . 6’ $ ‘ B
! R V
3 L9 L A “A 3 6# 7+ U مقدمه -1
\3 , OA [FGM !”#
S ‘ 9!+] L !”?A (‘ 3 !9
V FGM H & .[b [] A$ 9# [^_` . !39 P R dV’ dV’ %’ + f! 9R%’ + A F? 3 E9 e= 6 U J& f! 8 $ g C ; \3 FGM 0 R 9F . 6& !F
6′ O >A ; 6
9 N h’ L != E$ > \, 3 i G% @!, FGM ; ” . 9 9 = H & L 0″ A 9 %& ‘ 0 # ‘
-9Ej , . % 9N 9F A F
! FGM # J ! kA 9’ !> ) %* 8 !U 3
-S ‘ A “A L != .9 $ 0# ‘9%?3
6& ‘ FGM $ G%’ &
D BV > l%O’ 9N S B
‘9%?3 8 !U $j ; ?# (bHVAC)
%# E < 9” B% $ A ‘ 63 ,
.6′
! V L % 7E
$ “= 3 A %&+ FGM
\’ ! 9N C 37 V 3 L >A \ * 6OF E , $ ‘ f! ? 9R%O& kA $ + %# E U 3 i3 . 9 9F ! R!’
9 “A . o A o A .6′ $ =
E $ ‘ $ EF ( FG 9 V%O
K ‘ .6’ %# 7+ U [p] p6 (‘ Vm +Vc =1 ([)

6″O< 9″MN O3 \ Vm Vc L 3 $ = < 8 K ‘ . !9 $ ‘ 0 # 9F ‘
$ 4 [[t] [b + ‘ 3 9A
%’ ‘ 9″MN O3 P % + 6’ E9 ) U 6 *Q

z1 g
36288366778

Vc = ( + ) , g ≥ 0 (b) h2

h $ 6 *Q %’ J%* z L 3 L E |& g !s” . 9 L 6 *Q
-9 & GU O %E 0 3 6′ 9R- kA FGM 9F A F H & 9″ .E
) P % + K ‘ %O %’; S 6 *Q 3 L ‘ + \ Q !s” . A E9
3 6’ 3} T ; . 9 %# E kA 6 X X L ‘ + 6BOA P – 3 $
T MA V \ Y O ~’ + P 0 – A
FGM 4 O %’ $
L != ) A %# E kA %+ 9% X
K ‘ .( ! B [€€ /{•] 8, A “A
A9 ) U FG %+

E(z) =(Ec −Em )Vc (z)+Em ρ(z) =(ρc −ρm )Vc (z)+ρm (•)
ν(z)=ν

0 # H & \ c m |& ; V
$ ‘ $ = < ; V . !”A9 $ ‘
3 9< H & L9 L $ G%’ 3 9
. 6′ 6′ 0 # 9F ‘ 0,
‘ @!, 9!R” $ RA g= 0 F R! H & L E |& f 0# 6 l &
L E |& 3 9% N . ! 9 f 0# 0 # %’ 9V& U H & 6′ N
. “A9 P 6 *Q

1268732112263

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

9Q = 9′ [[b] z9E q!’
& .”A $ E
L % ? 9 M , 3 “A I # !- L @+ . !
9′ ! 3
[yK S i L != .6# 7+ U Z @A 3 # 6OA 0% . E F [[p] & 3 9V “, . 6′
3 L 9 6′ $ T MA 9′ [[`] {{ % 9′ B’
0% . $ G%’ P% 6 X 6 *Q $ ‘
.”A $
– ! V – E
%”3 9 6# 8 ! R”
$ %& ‘ 9′
.6′ $ %&+ FG !”#
0% /”, !- . E$ > f j+
\’ ! 9!%# 7+ 8
[email protected] 3 # 6′ %# E < V FG
Z F 9 BC
. 9 R” 9 @+ .6′ $ 6′
O l> %+ X $ 4 N 6 U
; N Z F 6′ $ 9′
/L%+ L% + /L% L% /L%
.6′ $ 4 L% + /L% + L%

2- مواد هدفمند FGM
-9 9%N 6& ‘ !# FGM $ L 9 9E ‘ >A H & P $ A !
\’ ! 9N C L 9 . “A S %!3 F “= S JN H & P R %>
L !3 .”A 9# L 6& ‘ 6>, $ ‘ \’ !
H & P 8 $A L 9G %* S
S %”> 9F .6′ $ 4 FG
. 9 9A $ ‘ S L (‘ $ 9#
$ %& ‘ 0 # ‘ \ 3 FG $ sA !-
6 l& $ RA

@A 3 # L 6′ 3 6′ A E
> , 8 – E . 9 T ; 9’ ‘ 9 M%A S @A 3 # 9 < x < @A 3 # r *%’ 9 A
3 6′ A ! . 9″A ; 9
3 A 3 $ G%’ \’ ! 9!%# 7+ 8
9 < @A 3 # L E$ > L %
$ 2= 6′ FG ) %* 9
. $ G%’ < ) %* ( L
$ 3} 9Ej ” 3 \’ ! 9? 8
6′ U

391839169788

x 2 y 2 k N i N j i j W = 2 + 2 −1 ∑∑αij x y (u)
ab i = =0 j 0

0 A k !9 S >M \Q αij ; V
( Q 3 6′ 9% X \ Q
-9 ) U 9 %# E F ) %* E

B Z : k = 0
$ ‘ $ E F Z : k =1 (w)
E B Z : k = 2

9?B!, ] A 3 6′ T ; 0% N k!
9?A 3 OA %+ ] A . A B’ OA %+
-9 ) U o A
E

1 a b h/2
=

∫ ∫ ∫ (σε σε τγx x + y y + xy xy )dxdydz
2− − −a b h/2
(_)

!%O fA 3 G γxy εy εx ; V
( L F P L G 7R , 3
!9 6′ U 9V& L F P /fA 3
∂2w∂2w εx =−z

2 , εx =−z

2 ,
1268732112263

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

∂x∂y (^)
∂2w γxy =−2z

∂ ∂x y
ورق بيضوي 3- تحليل ريتز براي ارتعاشات
FGM
93 A FGM F x V
b – 3 D a v 0 D 3 R kA
. 9 h 6 *Q

شكل 1- نمايي از يك ورق بيضوي

9 M , L ’23 1 K ‘ 9 %# E kA U

∂w
u(x, y,z,t) =−z

∂x
∂w
v(x, y,z,t) =−z

∂y
w(x, y,z,t) = w(x, y,t) (‚)

9!%# 7+ 8 %# 0% N T <
9 %’ \, 3 A 9 \’ ! % kA . x < S N = $ ‘ N
“* 9? 8 9 8
F L9 9A GU 9 M ,
6# E kA

54254690022

149047490022

=x22y22α1 x22 +α2 y22 +α3 (t)
 + −1
abab

-9 S >M \Q ‘ α3 α2 α1 ; V L 6’ F 8 [[y] O . !
$ ‘ ( @A 3 #
6 *Q 6BOA O “A $ G%’ r *%’ 9B’ ! ƒ , ) %* L ‘ + \Q L F? 9 A F P 8 . “A OA %+ ] A !? 3 L ?A L 9 & !-

z E z2 ( ) ∂2W 2 ∂2W 2
696646-11953

VMax =∫∫∫V 2(1−ν2)  ∂x 2  + ∂y 2  ([t)
∂2W ∂2W ∂2W 2
+2ν

22 +2(1−ν)

 dV 
∂x ∂y∂ ∂x y 

/ + S K ‘ FG ; V : 9 $ ‘ F
h3(g2 + +g 2)(E −E )
V


1∂2W ∂2W 
83516454332

+

Em ∫∫ 2  +2 
3 A  ∂x  ∂y 
∂2W W∂2 ∂2W 2
+2ν

22 +2(1−ν)

 dA
∂x ∂y∂ ∂x y   ({• )

jA E; 8 0% . E F k!
9 $ G%’ U

Π=V −T ([w)
22
MaxMax

%’ L 6′ T ; 0% N 9 %’ k!
] A >A 3 ! 6′ 9!%# 7+ 8 9! !”3 9F A F

δΠ=Π +(u δ δ δu v, + v w, + w )
([_)
−Π(u v w, ,) = 0

U 9? 8 3} 2B< 3 VA ” \ Q >! V .lE9 kA (u) V
“, !- B sA !- . 9 S >M αij
; V $ RA N i +N j 9? 8 6 l &

1268732112263

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

Downloaded from marine-eng.ir at 17:15 +0330 on Monday October 30th 2017

([^) /f! ( f! G V %#
U GU f! 6 N o fA 3
9 $ G%’
E(z)
σx =

2 (εx +νεy)
(1−ν )
365761132174

E(z) ([y ) σy =2 (εy +νεx)
(1−ν )
E(z)
τxy =

γxy
2(1+ν)

U 0 A o A 9?B!, ] A
9 )

179325-25328

h/2
Tρ(z)(w&)2 dxdydz ([[)
h/2

L 6BOA x%? ! w ; V A 6 2=
9?B!, ] A L9 ; V $ G%’ . 9
N . 6′ 9Q = 9 M , \ON
F 9Q = 9 M , 8 9 )

w(x, y,t) =W(x, y)sin(ωt) ([b )

@A 3 # ω $ 9? 8 W ; V
; V 7R , . 9 L ?A 9 BC
.3 B’ 9?B!, ] A L9 ([[) V
U 9?B!, ] A !? !
E9 B’

T

ρ( )z W dV2 ([p)
129540-236639

V

‘$ ‘ ; V (p) (b) ( 7R , FG 9?B!, ] A !?
:l ‘ 9 F /+ S K ‘

2  ρ ρc − m 2
TMax =



قیمت: تومان

دسته بندی : مهندسی دریا و بندر

دیدگاهتان را بنویسید