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Exact higher order solutions for a simple adaptive structure
~
Int. J. Solids Structures Vol. 35, Nos 26~ 27, pp. 3595 3610, 1998 © 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0020-7683/98 $19.00 + .00
Pergamon PII :
EXACT
S0020-7683(98)00024-9
HIGHER ORDER SOLUTIONS FOR SIMPLE ADAPTIVE STRUCTURE
A
PAOLO GAUDENZI* Universit/t di R o m a " L a Sapienza", Dipartimento Aerospaziale, Via Eudossiana 16, 00184 Rome, Italy
(Received 2 August 1997; in revisedform 23 December 1997) A b s t r a c t - - T h e analysis of an adaptive beam composed by a passive layer and two surface bonded induced strain actuators is considered. A simple higher order beam model is formulated and exact solutions are obtained for a case of m e m b r a n e actuation and for a pure bending case. The obtained solutions are then discussed in terms of the main geometrical parameters o f the system and compared with the classical closed form solutions based on Euler-Bernoulli models. As a result the interaction mechanism between the passive and the active part of the structure is better described t h a n in existing closed form models. Moreover the present model allows the description of the edge effect which occurs close to the free boundary of the considered structure. © 1998 Elsevier Science Ltd. All rights reserved.
1. I N T R O D U C T I O N
The analysis of intelligent structures composed by passive materials and active elements has recently received an increasing amount of interest (Crawley and de Louis, 1987; Crawley, 1994). Active materials such as piezoceramics (PZT) and piezopolymers (PVDF) are being studied as parts of vibration reduction mechanisms. Their role as sensors and actuators has been effectively demonstrated for simple feed-back control techniques both from an experimental and a theoretical point of view (Denoyer and Kwak, 1996 ; Gaudenzi et aL, 1997). An increasing number of modeling studies has recently proposed the analysis of homogeneous piezoelectric media. Gaudenzi and Bathe (1995) proposed a general finite element procedure for the nonlinear analysis of electroelastic materials. A closed form solution for rectangular plates was recently presented by Bisegna and Maceri (1996). Other studies consider the analysis of composite structures in which only a part of the system is made of active (piezoelectric) elements. In this case different constitutive relations are present in different regions such as in the finite element studies by Kim et al. (1997) and by Gaudenzi (1997). A number of contributions focus on the analysis of layered structures for which, starting from the work of Crawley and de Louis (1987), beam and plate models of increasing degree of complexity has been presented such as in the finite element works of Saravanos et al. (Saravanos and Heyliger, 1995 ; Saravanos et al., 1997) and in the case of closed form soliation proposed by Ray et al. (1993). In parallel with numerous existing studies on composite laminated structures (Reddy, 1987; Gaudenzi et al., 1995) higher order theories for active composite laminates can be proposed for improving the modeling capabilities of classical plate theories. In these studies, such as the ones by Saravanos and Hegyliger (1995), Saravanos et al. (1997) and by Barboni et al. (1995), the hypothesis of plane section is removed and more realistic kinematic models are proposed. In the present study exact solutions are presented for a higher order modeling approach of the behavior of an active cantilever beam. Two symmetrically surface bonded piezoelectric layers are present on the top and the bottom of a beam. A simple higher order kinematic is proposed which allows one to consider curved deformed configurations of the plane sections of the beam and more accurate simulation of the stress distributions. * Fax : + 39-6-44585670. E-mail : [email protected]. 3595
3596
P. Gaudenzi
The results, obtained both for a bending and a m e m b r a n e case, are compared with the ones obtained for the case of plane sections which remain normal to the axis o f the beam after deformation (Euler-Bernoulli case). The effects of the geometry and the material parameter on the proposed solutions are also shown and discussed in detail. 2. THEORETICAL FORMULATION The considered structure consists of two actuating piezoelectric layers perfectly bonded to a passive cantilever beam structure (Fig. 1). The linear response o f the system is studied in the bidimensional case. The structure is symmetric with respect to its middle plane and all its parts are governed by an isotropic elastic behavior. The piezoelectric parts exhibit a coupled piezo-elastic constitutive behavior in which the electric field is supposed to be given for each point of the material. Namely a constant electric field is assumed. F o r this reason the actuation mechanism is represented in the model simply by applied actuation strain components. In the following subsections the mathematical formulation of the problem is given and the governing equations in terms o f generalized unknown displacement are obtained. 2.1. K i n e m a t i c s
For the beam of Fig. 1 consisting o f two actuating layers o f thickness G and a passive structure of thickness ts the following form of displacement is assumed : u(x, z) = ~o(X) + ~,(x)z + ~ ( x ) z ~ + ~ ( x ) z ~ w(x, z) = Wo(X) + w ~(x)z + w2(x)z 2
(i)
which can be considered as a power series expansion o f the displacement components along the z-axis confined to a limited n u m b e r o f terms. In this way a segment normal to the middle plane of the structure m a y be assumed after deformation of a curved shape. As a consequence the expression of total strain components assumes the form, where [']' indicates d[']/dx :
~ = u;(x) + u', (x)z + ui(x)z ~ + u ; ( x ) ? ~ = W l(X) + 2 w i ( x ) z
Z
e-f.
h j
t~
T
wl, ~fj
>
>
< [. Fig. 1. Geometry of the beam.
×
Exact higher order solutions for a simple adaptive structure
3597
7= = u1(x) + w'o(x) + [2u2(x) + w'1(x)]z + [3u3(x) + w'z(x)]z 2
(2)
2.2. Constitutive relations Passive plate structure :
ax -
E, - - 2 ( ~ + v,8~) 1 -- V s
1 - - !2s
(3)
Zxz = GTxz Active piezoelectric layers :
E~ ---[(ex-Ai)q-Va(e~-A3)] 1 -v~ Ea
-- - - [ v o2 ( e . - A 1 ) q - ( e ~ - - A 3 ) ] 1 -
v~
(4)
GTxy
"Cxy ~ -
A1, A 3 are the normal actuation strain components in x and z direction, respectively. They can be considered as due to an electric field applied in the z direction that is the direction of polarization of the piezoelectric material. The pedix " s " denotes the passive structure while " a " identifies the actuators. 2.3. Governin9 equations The principle of virtual work is used to obtain the governing equations of the problem. In the absence of external applied loads the virtual internal load is equal to zero :
I"
(axbex + az&z + zxyfTxy) dz dx = 0 JO J-
(5)
(h/2)
where a unit width of the beam has been considered. By applying a virtual displacement 5Uo the only virtual deformation is rex = 6U'o and the preceding integral becomes :
f l f hI2 a x f u ; d z d x = 0
(6)
~(h/2)
By integrating by parts, the virtual displacement 6Uo appears instead o f its derivative with respect to x.
Ih/2
(v l'h/2 axdz6uo [~O-Jo J_
d - (h/2)
a'~fuo dx dz = O
(7)
(h/2)
The first term generates the natural boundary conditions. In the considered case since the first term has to vanish for every possible virtual displacement, either the normal force N x = Jfh/2 - ( h / 2 ) o-x • 1-dz or the axial displacement Uo are forced to be zero, which yields to the condition
3598
P. Gaudenzi
f l f h-l 2(hi2) a~(~u°dzdx=O,)
(8)
T h a t is :
flit t=12
Es
~-(t,12) E a
- - [ @ x2- m l ) - . l - v a ( G - A 3 ) ] d z - - 2 ('f'x-']- vsSz) dZ-'b- I dO LJ -(td2) 1 - v, d-(hl2) 1 -- Va ('hlZ E a -]t + [ T--~..z[(ex-Ai)+v~(~-A3)]dz[ dx = 0 d,,12 1 -- va ]
(9)
By integrating with respect to dz, the following differential e q u a t i o n is derived :
A IU~-t'- A3u~-'b BI w"1 = 0
(10)
With the corresponding b o u n d a r y conditions, respectively, obtained for x = 0, x = 1: Uo = 0
Aiu'o+Blwl-.l-A3u'2 = 0
(11)
AI, a3, B1 are constant terms better defined in the following. In the same way, by taking virtual variation o f the other six c o m p o n e n t s o f the displacement, the following set o f ordinary differential equations can be obtained :
Alu~--I- A3u~--l- Bxw'l = 0 A3u'~ - 4C3u2 --t-Asu~ + (B3 - 2C3)w~ = 0
BI u; + ( B 3 - - 2C3)u~ -['-A I w l -- C3 w'~ = L Im3m
(12)
C1 ul - A 3u'~-t- 3C3u3 - A 5u~ + Ci w~ q- (C 3 - 2B3)w ~ = 0 3C3ui -- A su'( -1- 9C5u3 -- A7u~ + 3C3w'o -.t-(3Cs -- 2Bs)w'2 = 0 Clu'l + 3C3u'3 + Cl w~ + C3w'~ = 0
(C3 -- 2B3)u~ + (3Cs -- 2Bs)u'3 + C3w; -- 4A3w2 -t--Csw'~ = 2L2A3b
(13)
With the b o u n d a r y conditions : x=0
x=l Alu'o + BlWl + A 3 u ~ - - L1AIm = 0
U0 ~ 0 U2 =
0
A3u'o-t- B3wl -b A s u ~ - Z3Aim = 0
wt~O
2C3u 2 --[-C3w"I = 0
ul = 0
A3u ~ + 2B3w2 + A s u - L z A I b = 0
U3=0
A sU'l -t'-2Bsw2 q- Ayu'3 -- L 4 A lb = 0
W0 = 0
f l U I -]-Ci wto--l-3C3u3--[-C3w'2 = 0
W2 ~-O
C3u I --}-C3w~o--l-3Csu3 -.]-Csw'2 = 0
Where the coefficients A~, Bk, Ck, Lk, and the actuating terms are :
(14)
Exact higher order solutions for a simple adaptive structure
5 F?-7_(',7] L~-kl-v~L\2) t,2)] h=2to+t, A~m =
m3m =
3599
(15)
(A~ + v~A~)+ (A~ + vaA~) 2
(A; + v~Ag)+ (A; + voA;) 2
Alb = (AT +vA~)-A~,,, (16)
A3b = (A; q-vaA~)--A3m
The superscripts u and l stand respectively for upper or lower actuator. As it can easily be seen, these equations can be grouped in two sets: the first one (first three equations) is relevant to a m e m b r a n e mode, the second one to a bending mode. Note that, since the dependence of the unknowns upon z has been established a priori, only total derivatives with respect to x appear in the formulation.
3. A CLOSED FORM SOLUTION FOR THE MEMBRANE MODE
(a) Let us now consider the m e m b r a n e m o d e and restrict the analysis in the case for which wt(x) = 0. The m e m b r a n e case is then governed by the following set of equations Alu'~'-I- A3u'~ = 0 A 3U~
--
4C3u2 -F A 5u~ --- 0
(17)
With the boundary conditions :
x = 0
~u0 = 0
t u2 x = l
0
~ AIu'°'bA3u'2 [A3u'o.+ A5u'2
L1Alm
LaAI~
(18)
The following solution form is assumed :
tu~(x)J from which the preceding system becomes :
tu2j
(19)
3600
P. Gaudenzi
Al~)2uo + A3(gZu2 = 0 A3~2Uo - 4 C 3 u 2 --I-As~b2u2 = 0
(20)
To obtain a n o n trivial solution, the characteristic p o l y n o m i a l o f this system has to vanish
O2[(A,A,--AZ)c/)Z--4C3A,] = 0
(21)
F o r which equations the solutions are : ~1,2 = 0
C3A1
1~3,4 = +2~I A 1 A s _ A ~ - +c~
(22)
The solution o f the system is therefore o f the f o r m :
U(x) = ~Uo(X)~ = Uz + U2x+ U3 e-~X+
(u2(x)J
U 4 d 'x
(23)
This solution must meet the equilibrium equations, for every value o f x, and the b o u n d a r y conditions. A 1(o~2uo3 e-~'x + (x2u04 e~X)+ A 3(odu23 e -~x + 0~2U24 e "x) = 0 A3(~2u03 e-~X+ °~2u04 d'x) -- 4C3(u21 + u22x~- u23 e - " x + U24 e=0
+As(~2u23 e-~XW~2u24 e ~x) = 0 By imposing that the sum o f the coefficients which a p p e a r multiplied by the same function of x vanishes, we obtain : Aluo3+A3u23 = 0
AlUo4+A3u24 = 0
/,/21 = 0
1d22= 0
(24)
and the conditions in x = 0, x = l: X=0
IU01 "~ U03 "{- U04 = 0 [.U21 -1- U23 "]- U24
x = l
0
~Al(u02 - ~ u 0 3 e-=l-t-o~u04 e=t)W A3(u22 -0~u23 e-~'t + ~u24 e~'~)= LiAlm [A3(uo2-ccuo3e-~l+~uo4e~t)+As(u22--o~uz3e-~l+c~u24e't)
(25)
L3Alm
Finally the solution can be written as : UOI ~---0 /'/02 ~ A~-AIm
A3
U03 ~ --U04 = -- Z U 2 3 U21 = U22 = 0 A3
U23 = --U24 =
\
A1
1
L3--~lL,)A,m-(A~_AsAl)e_~a-+e~t
(26)
Exact higher order solutions for a simple adaptive structure
3601
e ~x-- e-~x [ L1 ct A 3\ u ( x ' z ) = - ~ l A l m X + 4C3A, e - - L - ~ + ~ L 3 - L , - - ~ I ) A , , . ( - A a + A I Z 2)
(27)
(b) We n o w obtain the solution for the simple case for which the expression o f u(x, z) is limited to the constant term with respect to z :
(28)
u(x) = Uo(X)
T h e governing equation and the b o u n d a r y conditions are in this case :
Alu'~ = Ox = O uo = O x = l
Alu'o = L~Al,,
(29)
In conclusion the following solution is found :
U(X) -~ ~ AIm
(30)
Which can be recognized as the first term o b t a i n e d in the higher order case.
4. A CLOSED FORM SOLUTION FOR THE BENDING MODE (a) W e consider n o w the bending actuation case and restrict the analysis to the case for which the expression o f w(x, z) is limited to the c o n s t a n t term with respect to z, that is :
w(x) = Wo(X)
(31)
In this case the governing equations read :
C l u l - A3u'~ + 3C3u3- Asu~ + Clw'o = 0 3C3ul - A su'~ + 9C5u3 - hTU~ + 3C3w'o -- 0 Clu'l + 3C3u; + C1 w~; = 0
(32)
The assumed b o u n d a r y conditions are : ul = 0 x=0
Ju3=0 tWo = 0
x = 1
A 3u'l + A su'3-L2A1b = 0 IIAsU'l .-}-A7u'3-L4AIb = 0
(33)
/
tClU 1-.]-Cl Wto.+-3Cau3 ..~ 0
T h e following solution f o r m is a s s u m e d :
W ( x ) = l U a ( X ) l = u3 e *x
tw0(x))
tw0J
f r o m which the following h o m o g e n e o u s algebraic system is f o u n d :
(34)
P. Gaudenzi
3602
( C 1 -- A3(~2)Ul 71- ( 3 C 3 - A s ~ 2 ) u 3 -Jl-Cl~)W 0 = 0
(3C 3 - Asq~Z)Ul -I- ( 9 C , - - AT~b2)u3 + 3C3q~w0 = 0 (35)
CldgU 1 .q- 3C3~bu3 --]-Clq~2Wo = 0
By imposing the characteristics equation to vanish :
(a4[C,(A3A7 - A~)c~ 2 - 9A3(C, C5 - C32)] = 0
(36)
we obtain for ~b the following values : ~bi=0
i= 1,2,3,4
/A3(C, C~- C~)
(~1,2 = " ~
=
±3~]C~(A3A7_A2)
(37)
The solution o f the system is therefore o f the f o r m :
Iu,(x)] W(x) = lu3(x)/=
W l --[- W2x-~- W3x2 -~ W4x3 -~- Ws e-'X + W6 e ax
(38)
(Wo(X)i By substituting this expression in the governing equations we obtain : Cl(Ul i --~Ul2X-~- u I 3x2 -~-u14x3 + u15 e-~q.-ul6 e~X)--A3(2Ul3q-6Ul4Xq-a2ul5 e-~--~ o~2Ul6 e ~) + 3 C3 (u31 + u32x + u33x 2 + u3 4X3 + u35 e - ~ + u3 6 e°~x)- - A 5(2U33 "~ 6U34x q" ~2 U35 e - ' x + ~2u3 6 e~:~x) + CI(W02 "q-2Wo3X-'}-3Wo4x2 -- ~W05e-~-~+ ~w06 e ~'x) = 0 3C3(ul ~q- Ul 2 --I-ul 3x2 q- u14x3 31-Ul 5 e-Ctx-~ Ul 6 fax) _ As(2ul 3 -~ 6Ul 4X + ~2Ul 5 e-aX AT.~2Ul 6 fax) + 9 C5 (u3 j + u3 zx + u33x 2 + u34x 3 + u35 e - ~' + u3 6 e~XX) -- A 7(2U33 -'l-6U 34X q- 0~2U35 e - ~ + a2u36 e "x) x 3C 3l(wo2 -t- 2Wo3X--I-3Wo4x2 -- O~Wo5e - ~ + O~Wo6e ~) = 0 C1 (Ul 2 + 2ut 3x + 3Ul 4x 2 - otul 5 e - ~ + o~ul6e "x) + 3 C3(u 32 "~ 2U33X q- 3U34x2 -- ~U35 e-~,x
-]-o~u36e~)-k-Cl(2wo3q--6Wo4X.-l-o~2wose-~-':.-l-o~2Wo6e~x) = 0
(39)
These equations must be satisfied for every value o f x in the range 0 ~< x ~< l, that is the sum o f the coefficients which a p p e a r multiplied by the same function o f x must vanish. In this way a system o f twelve independent equations is f o u n d that, with the c o r r e s p o n d i n g b o u n d a r y conditions, can be solved to find the eighteen constant o f the problem.
u(x,z) = ~L2 3AlbXZ+
C l ° ~ e ~ X - e~- ~~' xL/ 4 - L 2
9A3(CI C5 - C 2)
w(x,z) = - - - -L2 A l b x 2 + C 1 A s - 3 C 3 A 3 2A3
eC'X+ e - ~ ' x - 2 ( L4--L 2~33) A~b
9A3(C1 C5 - C23)
A n d the curvature o f the axis k = - w" is :
~33) A , b ( - A s z + A 3 z 3)
e ' l + e ~/
(40)
(41)
Exact higher order solutions for a simple adaptive structure
LzCiAs-3C3A3e~Xh-e-~X[" w"(x) =
A3A~b t- CI(A3A7 - A 2 ) ~ e _ a l
3603
~33) LL4-Lz
mb
(42)
(b) We n o w consider the p r o b l e m in the case for which the expression o f the axial displacement is limited to the term Ul(X) :
u(x,z)=ul(x)z
(43)
therefore the governing equations read :
f l u 1 -A3u'[-l-Ciw~o .= 0
C~ u] + C~ w~ = 0
(44)
With the b o u n d a r y conditions :
x=0
~( Wui o== 00
x =l
~A3u'1-LxAIb = 0
(45)
(Clb/1 -t-ClW 0 = 0
A s s u m i n g again the following f o r m for the solution :
W(x)_=~ul(x)~=IUlle4~X
(46)
[Wo(X)J ~Wo~ and substituting it in the governing equations : C~ ul e ¢ x - q~2A3ul e4'X-t- dpCl Woe 4"x = 0
Cl4)ut e~X+ CI(~)2W0 e ~x = 0
(47)
The characteristic equation o f the system is : @4 = 0
(48)
W ( x ) = ~Ul(X) ~ = W 1.-Jr-W 2 x ...1-W3 x2 .2I- W4 x3
(49)
Then the solution can be written as :
[Wo(X)J
and the equilibrium equations b e c o m e : Cl (Ul 131-/-/12x -'1-Ul 3x2 AvUl 4x3 ) - A 3(2ul 3 + 6u 14x) + Cl (w02 --t-2w03x + 3Wo4x2) = 0
Cl (Ul 2 7!-2UI 3X + 3U14x2) -t- Cl (2Wo 3 + 6WoaX) = 0
(50)
These equations must be verified for every value o f x in the range 0 ~< x ~< l
ClUl1-2A3u13-t-Clwo2 = 0
ClUiz-6A3u14+2Clwo3
=
0
ClUl3--~-3flwo4 ~- 0
U14 ~--- 0
(51) While the b o u n d a r y conditions read :
oql j o u!~als le!Xe oql agpo aoJj oql j o XI!UIOIAoql UI ",Z JO sonl-eA lua.tOJjlp .toj o.mlonals oql JO $I.X13Oql ~tlO|l~ ~/t/ = 2 aoJ pzl+nleAO upzalS oql Jo uoDnq!alsIP oql soqposop ~ +srt~!d "uo!lnlos !IlnOUaOlJ-aoln~I oql ql!A~ £113eXO spuods0Jao3 uoIlnIos Jopao q~!q oql ogpo ooaj oql tuoxj Keaxe 'l~zql oos o1 olqIssod o n e s! 1I "polaodxo se 'oanlonJls oa!ssed oql pue saolen13e oql UOOAMOqoaeJaOlu+ oql o1 +u+puodsoJJo3 dtunf ~ s+q ~9 oql jo uo!lnq!Jlsl.p aql '.Ieln~oJ aae '~a pu~ n jo UO!lnq!.tls[p oql oI!qA~ "s!soqlodKq uo!loos ouNd oql j o os'eolaJ oql j o sloojja oql oos ol olq!ssod s! 1! 'tueoq oql jo soleu!paooa le!Xe oql j o sonl'eA luoaajjIp aoj u ~ o q s oae salq'~lJeA oql jo suo.~lnq[Jls.tp oql oaoqt~ 'soan~g asoql u I •palealsnll ! aae ([~tV(~a - I ) ] F g ol laadsaJ qlInX paz!IetUaou) ~9 ssaals Su!puaq aql j o pue ( ~ V uIeJls paanpu! aql ol laadsaa ql!n~ poz!IetUaou) ~a u!mls le!xe aql jo '(ase3 !IInouaala-Jaln~I aql Joj pau!elqo auo aql ol laadsaa ql!ta paz.tletmou ) n luatuaaNdsip pz!x~ oql j o ssau~IaIql aql ~uo[l~ uo!lnq!alsIp aql ~-g sg!d uI "paaapisuoa lsJg s! uo!lnlos aueaqtuatu oq& •osea aelno!la-ed qaea aoj paq!aasap se 'paanpoalu! aae lsaJalu! j o salqe!aen aql jo uo!lez!ieuaaou sno!ae A "saolenlae aql j o ssau)lo.tql aql pue tueoq aaAO[.Ilu130 aql j o qlguo I aql uoo~loq ~1/7 = a oIlea oql (!!) '. aolenlae aql jo ssamla!ql aql pue aanlanals aql j o ssau)D!ql aql uaatalaq "l/*l = ,L o!lea oql (.0 '. aJnlanals oMss-ed oql j o pue Ju3d saolenlae oql jo luotuo~uea.le oql oq.lJosop ol aopao I1.I tlosoq3 0.I13 SaOlOtrdeaed OAM puo s!ql o j+ "molsds oA.tldep'e aql JO ,Qlatl.lO0~ oql j o sttuol u! ~u!~ollOJ aql u! passnas!p aae uo!laas sno~Aaad aql u! pau!elqo suoIlnIos laexa aq& NOISSflDSI(I 'g
-uo!mlos Jop.lo lsalj oql slli.iol lsJg sl.t Ill su.n31uoo qo!qax 'uoIlnlos JapJo .toqg!q oql ql!ax luols!suoo s! llnsoa s!ql oseo gu!puaq oql u! o s l v (gg)
qv ~V r7 • s! ,,m -- = ~/oanleAan3 oql qo.~q~ moJ d
(~Tff)
ux'q I V
+vz _ = ( x > E7
zxqlv
~V = ~-~
(z'x)n
: oJ~ uo!13o~!p z 'x aql u.t luotua3~lds!p oql j o suotssoJdxa oq~, pu~ (¢;)
~V~ ~7
q l v - - -
0 ~
170M ~
~
E°dl4
~OM -~ IOM
qiv~V __-~7 0 =
t'Irt =
~[ n
gln =
lIn
: pou[elqo aae tuolqoad oql j o sluelsuoa lq~[o oql £IIeUId 0=(d
"P0at~+l £0AI ~+ ~0~1") D I + ( d l~I n+zl £I n+l ~I n+ I tn)t D ]
(z2)
q + g z " / = (zffln~+leinE+Zln)~v)~ l = x 0 = t o,~) 0 = t i n ~)
0=x
!zuapneD 'd
1709~
Exact higher order solutions for a simple adaptive structure
3605
1,1 0
1,08 1,06 C0
1,04 B
.....,
• ...,....
1,02
"'± •
-4
-3
o~._
-2
la B ~ ° I / a
,i
-1
0
1
2
3
4
Z
Fig. 2. Axial displacement (T = 5 ; r = 50 ; x = 0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
1,4 1,2 1 0,8 o
0,6
o
w#
0,4 "; •
I
I
t
-4
-3
-2
- - O f2-
,.j -1
O
°j-'
"".~)?2- ! . . . .
1
I 2
3
-0,4
Fig. 3. Axial strain (T = 5 ; r
=
50 ; x
=
0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
structure increases with respect to the E u l e r - B e r n o u l l i solution, as s h o w n also in Fig. 3. The extension o f this effect, which o n l y interest the edge o f the b e a m , d e p e n d s u p o n T ; the area interested to this effects increases with T. T h e pure b e n d i n g case is considered in Fig. 6 where the tip deflection o f the b e a m for several ratios r is s h o w n as a f u n c t i o n o f T. F r o m the picture it is possible to n o t e that significant differences with respect to the E u l e r - B e r n o u l l i m o d e l increase with T a n d a s s u m e larger values for low values o f r. I n Figs 7 a n d 8 the axial strain ( n o r m a l i z e d with respect to A1b) a n d the b e n d i n g stress ( n o r m a l i z e d with respect to E J[(1--•2)Alb]) are illustrated for r = 50 a n d T = 5 for several values o f x . I n a n a l o g y with the m e m b r a n e case the r e m o v a l o f plane section hypothesis is evident only in p r o x i m i t y o f the edge. F i g u r e 9 illustrates the i n d u c e d c u r v a t u r e ( n o r m a l i z e d with respect to 2Alb/ts) m e a s u r e d at x --- 0.95L for r = 50, as a f u n c t i o n o f T. Also in this case the differences b e t w e e n the
3606
P. Gaudenzi
0,6 "" f..o~
m -i
0,4 r
s I
I
I
'-I
I
0
-1 " ' ~ . . 1 ..-''1 -0,2 ~ "
-2
•
3,"
2
s
-0,4 -0,6 ..0,8
Fig, 4. Axial stress (T = 5 ; r = 50 ; x = 0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
.i
J#
;t o## m
• ~,w#~ f.
1 0 0
I
I
I
I
I
I
0,2
0,4
0,6
0,8
1
1,2
~L
Fig. 5. Axial strain (z = hi2 ; r = 50 ; T = 5 solid line ; T = 15 dashed line ; T = 30 dotted line).
E u l e r - B e r n o u l l i s o l u t i o n a n d the p r e s e n t one are i m p o r t a n t o n l y as T b e c o m e s large. T h e " P i n - f o r c e " results (1) are also s h o w n for c o m p a r i s o n . I n Fig. 10 the c u r v a t u r e is n o r m a l i z e d w i t h respect to A~b. E u l e r - B e r n o u l l i (solid line) a n d high o r d e r s o l u t i o n s h o w a r e l e v a n t difference o n l y as T increases as b e t t e r i l l u s t r a t e d in Fig. 11 w h e r e the c u r v a t u r e is p l o t t e d s t a r t i n g f r o m T = 5. F o r low values o f T the higher o r d e r a n d the E u l e r - B e r n o u l l i m o d e l s give the s a m e results. F r o m Fig. 12, in w h i c h the c u r v a t u r e is n o r m a l i z e d w i t h respect to the c o r r e s p o n d i n g E u l e r - B e r n o u U i solution, the e x t e n s i o n o f the edge effect for different values o f T c a n be e x a m i n e d . T h e larger is T the m o r e e x t e n d e d is the p o r t i o n o f the b e a m interested b y the effect. A s r a n d L are fixed (in this case L = 50, r = 50) the effect o f the increase o f T c a n be i n t e r p r e t e d as a shear d e f o r m a t i o n effect, r e p r e s e n t e d b y h i g h e r o r d e r t e r m s o f the p o w e r e x p a n s i o n o f d i s p l a c e m e n t c o m p o n e n t s in t e r m s o f z.
Exact higher order solutions for a simple adaptive structure
3607
1,35 1,3 1,25
t
1,2 1,15 1,1
1,05 1m 0
•_--..--..-'+-..-.'7."..-. . . . . 10
I" " r " " . . . . . . . 15
] 20
T Fig. 6. Normalized deflection (x = L; r = 10 solid line; r = 25 dashed line; r = 30 dotted line).
it
0,5 [ I
-4
-3 eI
J e w
.
''
*'T
. . . . .
iI
.rA
.°:2 -0,
1
2
3
4
15
Fig. 7. Axial strain (T = 5 ; r = 50 ; x = 0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
6. CONCLUSIONS T h e p r o p o s e d m o d e l , a l t h o u g h very simple f r o m the p o i n t o f view o f the f o r m u l a t i o n , h a s p e r m i t t e d one to e v a l u a t e the i m p o r t a n c e o f h i g h e r o r d e r effects in the m o d e l i n g o f the i n t e r a c t i o n between i n d u c e d strain a c t u a t o r s a n d a simple b e a m structure. T h e closed f o r m s o l u t i o n o b t a i n e d for the m e m b r a n e a n d b e n d i n g cases has a l l o w e d o n e to identify the m a i n g e o m e t r i c a l a n d m a t e r i a l p a r a m e t e r s o f the p r o b l e m a n d their role in the r e s p o n s e o f the system. By e x a m i n i n g the results o b t a i n e d with this m o d e l , the r a n g e o f a p p l i c a b i l i t y o f the classical E u l e r - B e r n o u l l i m o d e l are b e t t e r assessed. M o r e o v e r the m o d e l p e r m i t s one to p r e d i c t the extension o f the a r e a interested b y the edge effect. T h e m e c h a n i c a l i n t e r a c t i o n b e t w e e n surface b o n d e d i n d u c e d strain a c t u a t o r s a n d passive s t r u c t u r e is e v a l u a t e d in m o r e detail t h a n with classical b e a m m o d e l with p o s s i b l e a d v a n t a g e s for the design process.
3608
P. G a u d e n z i
.:
o,s 0,4
.
:
-#
:"
0.2
,'"
,
: -4
":
~i
.0.6-1-
.,
F i g . 8. A x i a l s t r e s s ( T = 5 ; r = 50 ; x = 0 . 9 L s o l i d line ; x = 0 . 9 5 L d a s h e d line ; x = L d o t t e d line).
1,~ 1 "b
0,8 "
~
0,6 0,4.
0,20
I
I
I
I
5
10
15
20
T
F i g . 9. N o r m a l i z e d
c u r v a t u r e Kt,/2A~b (r = 50 ; x = 0 . 9 5 L ; p i n - f o r c e : d o t t e d line ; E u l e r - B e r n o u l l i s o l i d line ; h i g h e r o r d e r : d a s h e d line).
:
1,5
1,2 0,9 0,6 0,3 0 0
F i g . 10. N o r m a l i z e d
I
I
I
I
I
I
3
6
9
12
15
18
c u r v a t u r e K/Ajb (r = 50 ; x = 0 . 9 5 L ; E u l e r - B e r n o u l l i d o t t e d line).
: s o l i d line ; h i g h e r o r d e r :
Exact higher order solutions for a simple adaptive structure
3609
0,3
°,,t
~0,2
0.1 ]
-,
0 5
I
I
I
I
I
8
11
14
17
20
Fig. 11. Normalized curvature K/Alb (r = 50 ; x = 0.95L ; Euler-Bernoulli : solid line ; higher order: dotted line).
3,5 3
;I "1
2,5. "
,r,=
;i
2
=## ,~
°°° ,IJ
1,5 1 0,5 0 0
~2
~4
~6
0,8
1
x/I
Fig. 12. Normalized curvature K/Keb (r = 50; T = 5 solid line; T = 15 dashed line; T = 30 dotted line).
Acknowledgement--The financial support of the research project "Controllo delle vibrazioni di semplici strutture intelligenti" (MURST- Italian Ministry for University and Research--Rif. 410/1995) is gratefully acknowledged.
REFERENCES Barboni, R., Gaudenzi, P. and Strambi, G. (1995) On the modeling of actuator-structure interaction in intelligent structures. Intelligent Materials and Systems, Techna, pp. 291-298. Bisegna, P. and Maceri, F. (1996) An exact three-dimensional solution for simply supported rectangular piezoelectric plates. A S M E Journal of Applied Mechanics 63, 628~638. Crawley, E. F. (1994) Intelligent structures for aerospace: a technology overview and assessment. AIAA Journal 32(8), 1689-1699. Crawley, E. F. and de Louis, J. (1987) Use of piezoelectric actuators as element of intelligent structures. A1AA Journal 25(10), 1373-1385. Denoyer, K. K. and Kwak, M. K. (1996) Dynamics modeling and vibration suppression of a slewing structure utilizing piezoelectric sensors and actuators. Journal of Soundand Vibration 189(1), 13-31. Gaudenzi, P. (1997) On the electromechanical response of active composite materials with piezoelectric inclusion. Computers and Structures 65(2), 157-168. Gaudenzi, P. and Bathe, K. J. (1995) An iterative finite element procedure for the analysis of piezoelectric continua. Journal of lntelligent Material System and Structures 6, 266-273.
3610
P. Gaudenzi
Gaudenzi, P., Barboni, R. and Mannini, A. (1995) A finite element evaluation of single-layer and multi-layer theories for the analysis of laminated plates. Composite Structures 30, 427~,40. Gaudenzi, P., Barboni, R., Carbonaro, R. and Accettella, S. (1997) Direct position and velocity feed-back control on an active beam with PZT sensors and actuators. International Forum on Aeroelasticity and Structural Dynamics, 17-20 June 1997, Vol. II, pp. 285-292 Rome, CEAS. Kim, J., Varadan, V. V. and Varadan, V. K. (1997) Finite element modeling of structures including piezoelectric active devices. International Journal for Numerical Methods in En#ineering 40, 817-832. Ray, M. C., Rao, K. M. and Samanta, B. (1993) Exact solution for static analysis of intelligent structures under cylindrical bending. Computers and Structures 47(6), 1031-1042. Reddy, J. N. (1987) A generalization of two-dimensional theories of laminated composite plates. Common Appl. Numer. Meth. 3, 173-181. Saravanos, D. A. and Heyliger, P. R. (1995) Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. Journal of lntelli#ent Material System and Structures 6, 350-363. Saravanos, D. A., Heyliger, P. R. and Hopkins, D. A. (1997) Layerwise mechanics and finite element for the dynamics analysis of piezoelectric composite plates. International Journal of Solids and Structures 34, 359-378.
Int. J. Solids Structures Vol. 35, Nos 26~ 27, pp. 3595 3610, 1998 © 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0020-7683/98 $19.00 + .00
Pergamon PII :
EXACT
S0020-7683(98)00024-9
HIGHER ORDER SOLUTIONS FOR SIMPLE ADAPTIVE STRUCTURE
A
PAOLO GAUDENZI* Universit/t di R o m a " L a Sapienza", Dipartimento Aerospaziale, Via Eudossiana 16, 00184 Rome, Italy
(Received 2 August 1997; in revisedform 23 December 1997) A b s t r a c t - - T h e analysis of an adaptive beam composed by a passive layer and two surface bonded induced strain actuators is considered. A simple higher order beam model is formulated and exact solutions are obtained for a case of m e m b r a n e actuation and for a pure bending case. The obtained solutions are then discussed in terms of the main geometrical parameters o f the system and compared with the classical closed form solutions based on Euler-Bernoulli models. As a result the interaction mechanism between the passive and the active part of the structure is better described t h a n in existing closed form models. Moreover the present model allows the description of the edge effect which occurs close to the free boundary of the considered structure. © 1998 Elsevier Science Ltd. All rights reserved.
1. I N T R O D U C T I O N
The analysis of intelligent structures composed by passive materials and active elements has recently received an increasing amount of interest (Crawley and de Louis, 1987; Crawley, 1994). Active materials such as piezoceramics (PZT) and piezopolymers (PVDF) are being studied as parts of vibration reduction mechanisms. Their role as sensors and actuators has been effectively demonstrated for simple feed-back control techniques both from an experimental and a theoretical point of view (Denoyer and Kwak, 1996 ; Gaudenzi et aL, 1997). An increasing number of modeling studies has recently proposed the analysis of homogeneous piezoelectric media. Gaudenzi and Bathe (1995) proposed a general finite element procedure for the nonlinear analysis of electroelastic materials. A closed form solution for rectangular plates was recently presented by Bisegna and Maceri (1996). Other studies consider the analysis of composite structures in which only a part of the system is made of active (piezoelectric) elements. In this case different constitutive relations are present in different regions such as in the finite element studies by Kim et al. (1997) and by Gaudenzi (1997). A number of contributions focus on the analysis of layered structures for which, starting from the work of Crawley and de Louis (1987), beam and plate models of increasing degree of complexity has been presented such as in the finite element works of Saravanos et al. (Saravanos and Heyliger, 1995 ; Saravanos et al., 1997) and in the case of closed form soliation proposed by Ray et al. (1993). In parallel with numerous existing studies on composite laminated structures (Reddy, 1987; Gaudenzi et al., 1995) higher order theories for active composite laminates can be proposed for improving the modeling capabilities of classical plate theories. In these studies, such as the ones by Saravanos and Hegyliger (1995), Saravanos et al. (1997) and by Barboni et al. (1995), the hypothesis of plane section is removed and more realistic kinematic models are proposed. In the present study exact solutions are presented for a higher order modeling approach of the behavior of an active cantilever beam. Two symmetrically surface bonded piezoelectric layers are present on the top and the bottom of a beam. A simple higher order kinematic is proposed which allows one to consider curved deformed configurations of the plane sections of the beam and more accurate simulation of the stress distributions. * Fax : + 39-6-44585670. E-mail : [email protected]. 3595
3596
P. Gaudenzi
The results, obtained both for a bending and a m e m b r a n e case, are compared with the ones obtained for the case of plane sections which remain normal to the axis o f the beam after deformation (Euler-Bernoulli case). The effects of the geometry and the material parameter on the proposed solutions are also shown and discussed in detail. 2. THEORETICAL FORMULATION The considered structure consists of two actuating piezoelectric layers perfectly bonded to a passive cantilever beam structure (Fig. 1). The linear response o f the system is studied in the bidimensional case. The structure is symmetric with respect to its middle plane and all its parts are governed by an isotropic elastic behavior. The piezoelectric parts exhibit a coupled piezo-elastic constitutive behavior in which the electric field is supposed to be given for each point of the material. Namely a constant electric field is assumed. F o r this reason the actuation mechanism is represented in the model simply by applied actuation strain components. In the following subsections the mathematical formulation of the problem is given and the governing equations in terms o f generalized unknown displacement are obtained. 2.1. K i n e m a t i c s
For the beam of Fig. 1 consisting o f two actuating layers o f thickness G and a passive structure of thickness ts the following form of displacement is assumed : u(x, z) = ~o(X) + ~,(x)z + ~ ( x ) z ~ + ~ ( x ) z ~ w(x, z) = Wo(X) + w ~(x)z + w2(x)z 2
(i)
which can be considered as a power series expansion o f the displacement components along the z-axis confined to a limited n u m b e r o f terms. In this way a segment normal to the middle plane of the structure m a y be assumed after deformation of a curved shape. As a consequence the expression of total strain components assumes the form, where [']' indicates d[']/dx :
~ = u;(x) + u', (x)z + ui(x)z ~ + u ; ( x ) ? ~ = W l(X) + 2 w i ( x ) z
Z
e-f.
h j
t~
T
wl, ~fj
>
>
< [. Fig. 1. Geometry of the beam.
×
Exact higher order solutions for a simple adaptive structure
3597
7= = u1(x) + w'o(x) + [2u2(x) + w'1(x)]z + [3u3(x) + w'z(x)]z 2
(2)
2.2. Constitutive relations Passive plate structure :
ax -
E, - - 2 ( ~ + v,8~) 1 -- V s
1 - - !2s
(3)
Zxz = GTxz Active piezoelectric layers :
E~ ---[(ex-Ai)q-Va(e~-A3)] 1 -v~ Ea
-- - - [ v o2 ( e . - A 1 ) q - ( e ~ - - A 3 ) ] 1 -
v~
(4)
GTxy
"Cxy ~ -
A1, A 3 are the normal actuation strain components in x and z direction, respectively. They can be considered as due to an electric field applied in the z direction that is the direction of polarization of the piezoelectric material. The pedix " s " denotes the passive structure while " a " identifies the actuators. 2.3. Governin9 equations The principle of virtual work is used to obtain the governing equations of the problem. In the absence of external applied loads the virtual internal load is equal to zero :
I"
(axbex + az&z + zxyfTxy) dz dx = 0 JO J-
(5)
(h/2)
where a unit width of the beam has been considered. By applying a virtual displacement 5Uo the only virtual deformation is rex = 6U'o and the preceding integral becomes :
f l f hI2 a x f u ; d z d x = 0
(6)
~(h/2)
By integrating by parts, the virtual displacement 6Uo appears instead o f its derivative with respect to x.
Ih/2
(v l'h/2 axdz6uo [~O-Jo J_
d - (h/2)
a'~fuo dx dz = O
(7)
(h/2)
The first term generates the natural boundary conditions. In the considered case since the first term has to vanish for every possible virtual displacement, either the normal force N x = Jfh/2 - ( h / 2 ) o-x • 1-dz or the axial displacement Uo are forced to be zero, which yields to the condition
3598
P. Gaudenzi
f l f h-l 2(hi2) a~(~u°dzdx=O,)
(8)
T h a t is :
flit t=12
Es
~-(t,12) E a
- - [ @ x2- m l ) - . l - v a ( G - A 3 ) ] d z - - 2 ('f'x-']- vsSz) dZ-'b- I dO LJ -(td2) 1 - v, d-(hl2) 1 -- Va ('hlZ E a -]t + [ T--~..z[(ex-Ai)+v~(~-A3)]dz[ dx = 0 d,,12 1 -- va ]
(9)
By integrating with respect to dz, the following differential e q u a t i o n is derived :
A IU~-t'- A3u~-'b BI w"1 = 0
(10)
With the corresponding b o u n d a r y conditions, respectively, obtained for x = 0, x = 1: Uo = 0
Aiu'o+Blwl-.l-A3u'2 = 0
(11)
AI, a3, B1 are constant terms better defined in the following. In the same way, by taking virtual variation o f the other six c o m p o n e n t s o f the displacement, the following set o f ordinary differential equations can be obtained :
Alu~--I- A3u~--l- Bxw'l = 0 A3u'~ - 4C3u2 --t-Asu~ + (B3 - 2C3)w~ = 0
BI u; + ( B 3 - - 2C3)u~ -['-A I w l -- C3 w'~ = L Im3m
(12)
C1 ul - A 3u'~-t- 3C3u3 - A 5u~ + Ci w~ q- (C 3 - 2B3)w ~ = 0 3C3ui -- A su'( -1- 9C5u3 -- A7u~ + 3C3w'o -.t-(3Cs -- 2Bs)w'2 = 0 Clu'l + 3C3u'3 + Cl w~ + C3w'~ = 0
(C3 -- 2B3)u~ + (3Cs -- 2Bs)u'3 + C3w; -- 4A3w2 -t--Csw'~ = 2L2A3b
(13)
With the b o u n d a r y conditions : x=0
x=l Alu'o + BlWl + A 3 u ~ - - L1AIm = 0
U0 ~ 0 U2 =
0
A3u'o-t- B3wl -b A s u ~ - Z3Aim = 0
wt~O
2C3u 2 --[-C3w"I = 0
ul = 0
A3u ~ + 2B3w2 + A s u - L z A I b = 0
U3=0
A sU'l -t'-2Bsw2 q- Ayu'3 -- L 4 A lb = 0
W0 = 0
f l U I -]-Ci wto--l-3C3u3--[-C3w'2 = 0
W2 ~-O
C3u I --}-C3w~o--l-3Csu3 -.]-Csw'2 = 0
Where the coefficients A~, Bk, Ck, Lk, and the actuating terms are :
(14)
Exact higher order solutions for a simple adaptive structure
5 F?-7_(',7] L~-kl-v~L\2) t,2)] h=2to+t, A~m =
m3m =
3599
(15)
(A~ + v~A~)+ (A~ + vaA~) 2
(A; + v~Ag)+ (A; + voA;) 2
Alb = (AT +vA~)-A~,,, (16)
A3b = (A; q-vaA~)--A3m
The superscripts u and l stand respectively for upper or lower actuator. As it can easily be seen, these equations can be grouped in two sets: the first one (first three equations) is relevant to a m e m b r a n e mode, the second one to a bending mode. Note that, since the dependence of the unknowns upon z has been established a priori, only total derivatives with respect to x appear in the formulation.
3. A CLOSED FORM SOLUTION FOR THE MEMBRANE MODE
(a) Let us now consider the m e m b r a n e m o d e and restrict the analysis in the case for which wt(x) = 0. The m e m b r a n e case is then governed by the following set of equations Alu'~'-I- A3u'~ = 0 A 3U~
--
4C3u2 -F A 5u~ --- 0
(17)
With the boundary conditions :
x = 0
~u0 = 0
t u2 x = l
0
~ AIu'°'bA3u'2 [A3u'o.+ A5u'2
L1Alm
LaAI~
(18)
The following solution form is assumed :
tu~(x)J from which the preceding system becomes :
tu2j
(19)
3600
P. Gaudenzi
Al~)2uo + A3(gZu2 = 0 A3~2Uo - 4 C 3 u 2 --I-As~b2u2 = 0
(20)
To obtain a n o n trivial solution, the characteristic p o l y n o m i a l o f this system has to vanish
O2[(A,A,--AZ)c/)Z--4C3A,] = 0
(21)
F o r which equations the solutions are : ~1,2 = 0
C3A1
1~3,4 = +2~I A 1 A s _ A ~ - +c~
(22)
The solution o f the system is therefore o f the f o r m :
U(x) = ~Uo(X)~ = Uz + U2x+ U3 e-~X+
(u2(x)J
U 4 d 'x
(23)
This solution must meet the equilibrium equations, for every value o f x, and the b o u n d a r y conditions. A 1(o~2uo3 e-~'x + (x2u04 e~X)+ A 3(odu23 e -~x + 0~2U24 e "x) = 0 A3(~2u03 e-~X+ °~2u04 d'x) -- 4C3(u21 + u22x~- u23 e - " x + U24 e=0
+As(~2u23 e-~XW~2u24 e ~x) = 0 By imposing that the sum o f the coefficients which a p p e a r multiplied by the same function of x vanishes, we obtain : Aluo3+A3u23 = 0
AlUo4+A3u24 = 0
/,/21 = 0
1d22= 0
(24)
and the conditions in x = 0, x = l: X=0
IU01 "~ U03 "{- U04 = 0 [.U21 -1- U23 "]- U24
x = l
0
~Al(u02 - ~ u 0 3 e-=l-t-o~u04 e=t)W A3(u22 -0~u23 e-~'t + ~u24 e~'~)= LiAlm [A3(uo2-ccuo3e-~l+~uo4e~t)+As(u22--o~uz3e-~l+c~u24e't)
(25)
L3Alm
Finally the solution can be written as : UOI ~---0 /'/02 ~ A~-AIm
A3
U03 ~ --U04 = -- Z U 2 3 U21 = U22 = 0 A3
U23 = --U24 =
\
A1
1
L3--~lL,)A,m-(A~_AsAl)e_~a-+e~t
(26)
Exact higher order solutions for a simple adaptive structure
3601
e ~x-- e-~x [ L1 ct A 3\ u ( x ' z ) = - ~ l A l m X + 4C3A, e - - L - ~ + ~ L 3 - L , - - ~ I ) A , , . ( - A a + A I Z 2)
(27)
(b) We n o w obtain the solution for the simple case for which the expression o f u(x, z) is limited to the constant term with respect to z :
(28)
u(x) = Uo(X)
T h e governing equation and the b o u n d a r y conditions are in this case :
Alu'~ = Ox = O uo = O x = l
Alu'o = L~Al,,
(29)
In conclusion the following solution is found :
U(X) -~ ~ AIm
(30)
Which can be recognized as the first term o b t a i n e d in the higher order case.
4. A CLOSED FORM SOLUTION FOR THE BENDING MODE (a) W e consider n o w the bending actuation case and restrict the analysis to the case for which the expression o f w(x, z) is limited to the c o n s t a n t term with respect to z, that is :
w(x) = Wo(X)
(31)
In this case the governing equations read :
C l u l - A3u'~ + 3C3u3- Asu~ + Clw'o = 0 3C3ul - A su'~ + 9C5u3 - hTU~ + 3C3w'o -- 0 Clu'l + 3C3u; + C1 w~; = 0
(32)
The assumed b o u n d a r y conditions are : ul = 0 x=0
Ju3=0 tWo = 0
x = 1
A 3u'l + A su'3-L2A1b = 0 IIAsU'l .-}-A7u'3-L4AIb = 0
(33)
/
tClU 1-.]-Cl Wto.+-3Cau3 ..~ 0
T h e following solution f o r m is a s s u m e d :
W ( x ) = l U a ( X ) l = u3 e *x
tw0(x))
tw0J
f r o m which the following h o m o g e n e o u s algebraic system is f o u n d :
(34)
P. Gaudenzi
3602
( C 1 -- A3(~2)Ul 71- ( 3 C 3 - A s ~ 2 ) u 3 -Jl-Cl~)W 0 = 0
(3C 3 - Asq~Z)Ul -I- ( 9 C , - - AT~b2)u3 + 3C3q~w0 = 0 (35)
CldgU 1 .q- 3C3~bu3 --]-Clq~2Wo = 0
By imposing the characteristics equation to vanish :
(a4[C,(A3A7 - A~)c~ 2 - 9A3(C, C5 - C32)] = 0
(36)
we obtain for ~b the following values : ~bi=0
i= 1,2,3,4
/A3(C, C~- C~)
(~1,2 = " ~
=
±3~]C~(A3A7_A2)
(37)
The solution o f the system is therefore o f the f o r m :
Iu,(x)] W(x) = lu3(x)/=
W l --[- W2x-~- W3x2 -~ W4x3 -~- Ws e-'X + W6 e ax
(38)
(Wo(X)i By substituting this expression in the governing equations we obtain : Cl(Ul i --~Ul2X-~- u I 3x2 -~-u14x3 + u15 e-~q.-ul6 e~X)--A3(2Ul3q-6Ul4Xq-a2ul5 e-~--~ o~2Ul6 e ~) + 3 C3 (u31 + u32x + u33x 2 + u3 4X3 + u35 e - ~ + u3 6 e°~x)- - A 5(2U33 "~ 6U34x q" ~2 U35 e - ' x + ~2u3 6 e~:~x) + CI(W02 "q-2Wo3X-'}-3Wo4x2 -- ~W05e-~-~+ ~w06 e ~'x) = 0 3C3(ul ~q- Ul 2 --I-ul 3x2 q- u14x3 31-Ul 5 e-Ctx-~ Ul 6 fax) _ As(2ul 3 -~ 6Ul 4X + ~2Ul 5 e-aX AT.~2Ul 6 fax) + 9 C5 (u3 j + u3 zx + u33x 2 + u34x 3 + u35 e - ~' + u3 6 e~XX) -- A 7(2U33 -'l-6U 34X q- 0~2U35 e - ~ + a2u36 e "x) x 3C 3l(wo2 -t- 2Wo3X--I-3Wo4x2 -- O~Wo5e - ~ + O~Wo6e ~) = 0 C1 (Ul 2 + 2ut 3x + 3Ul 4x 2 - otul 5 e - ~ + o~ul6e "x) + 3 C3(u 32 "~ 2U33X q- 3U34x2 -- ~U35 e-~,x
-]-o~u36e~)-k-Cl(2wo3q--6Wo4X.-l-o~2wose-~-':.-l-o~2Wo6e~x) = 0
(39)
These equations must be satisfied for every value o f x in the range 0 ~< x ~< l, that is the sum o f the coefficients which a p p e a r multiplied by the same function o f x must vanish. In this way a system o f twelve independent equations is f o u n d that, with the c o r r e s p o n d i n g b o u n d a r y conditions, can be solved to find the eighteen constant o f the problem.
u(x,z) = ~L2 3AlbXZ+
C l ° ~ e ~ X - e~- ~~' xL/ 4 - L 2
9A3(CI C5 - C 2)
w(x,z) = - - - -L2 A l b x 2 + C 1 A s - 3 C 3 A 3 2A3
eC'X+ e - ~ ' x - 2 ( L4--L 2~33) A~b
9A3(C1 C5 - C23)
A n d the curvature o f the axis k = - w" is :
~33) A , b ( - A s z + A 3 z 3)
e ' l + e ~/
(40)
(41)
Exact higher order solutions for a simple adaptive structure
LzCiAs-3C3A3e~Xh-e-~X[" w"(x) =
A3A~b t- CI(A3A7 - A 2 ) ~ e _ a l
3603
~33) LL4-Lz
mb
(42)
(b) We n o w consider the p r o b l e m in the case for which the expression o f the axial displacement is limited to the term Ul(X) :
u(x,z)=ul(x)z
(43)
therefore the governing equations read :
f l u 1 -A3u'[-l-Ciw~o .= 0
C~ u] + C~ w~ = 0
(44)
With the b o u n d a r y conditions :
x=0
~( Wui o== 00
x =l
~A3u'1-LxAIb = 0
(45)
(Clb/1 -t-ClW 0 = 0
A s s u m i n g again the following f o r m for the solution :
W(x)_=~ul(x)~=IUlle4~X
(46)
[Wo(X)J ~Wo~ and substituting it in the governing equations : C~ ul e ¢ x - q~2A3ul e4'X-t- dpCl Woe 4"x = 0
Cl4)ut e~X+ CI(~)2W0 e ~x = 0
(47)
The characteristic equation o f the system is : @4 = 0
(48)
W ( x ) = ~Ul(X) ~ = W 1.-Jr-W 2 x ...1-W3 x2 .2I- W4 x3
(49)
Then the solution can be written as :
[Wo(X)J
and the equilibrium equations b e c o m e : Cl (Ul 131-/-/12x -'1-Ul 3x2 AvUl 4x3 ) - A 3(2ul 3 + 6u 14x) + Cl (w02 --t-2w03x + 3Wo4x2) = 0
Cl (Ul 2 7!-2UI 3X + 3U14x2) -t- Cl (2Wo 3 + 6WoaX) = 0
(50)
These equations must be verified for every value o f x in the range 0 ~< x ~< l
ClUl1-2A3u13-t-Clwo2 = 0
ClUiz-6A3u14+2Clwo3
=
0
ClUl3--~-3flwo4 ~- 0
U14 ~--- 0
(51) While the b o u n d a r y conditions read :
oql j o u!~als le!Xe oql agpo aoJj oql j o XI!UIOIAoql UI ",Z JO sonl-eA lua.tOJjlp .toj o.mlonals oql JO $I.X13Oql ~tlO|l~ ~/t/ = 2 aoJ pzl+nleAO upzalS oql Jo uoDnq!alsIP oql soqposop ~ +srt~!d "uo!lnlos !IlnOUaOlJ-aoln~I oql ql!A~ £113eXO spuods0Jao3 uoIlnIos Jopao q~!q oql ogpo ooaj oql tuoxj Keaxe 'l~zql oos o1 olqIssod o n e s! 1I "polaodxo se 'oanlonJls oa!ssed oql pue saolen13e oql UOOAMOqoaeJaOlu+ oql o1 +u+puodsoJJo3 dtunf ~ s+q ~9 oql jo uo!lnq!Jlsl.p aql '.Ieln~oJ aae '~a pu~ n jo UO!lnq!.tls[p oql oI!qA~ "s!soqlodKq uo!loos ouNd oql j o os'eolaJ oql j o sloojja oql oos ol olq!ssod s! 1! 'tueoq oql jo soleu!paooa le!Xe oql j o sonl'eA luoaajjIp aoj u ~ o q s oae salq'~lJeA oql jo suo.~lnq[Jls.tp oql oaoqt~ 'soan~g asoql u I •palealsnll ! aae ([~tV(~a - I ) ] F g ol laadsaJ qlInX paz!IetUaou) ~9 ssaals Su!puaq aql j o pue ( ~ V uIeJls paanpu! aql ol laadsaa ql!n~ poz!IetUaou) ~a u!mls le!xe aql jo '(ase3 !IInouaala-Jaln~I aql Joj pau!elqo auo aql ol laadsaa ql!ta paz.tletmou ) n luatuaaNdsip pz!x~ oql j o ssau~IaIql aql ~uo[l~ uo!lnq!alsIp aql ~-g sg!d uI "paaapisuoa lsJg s! uo!lnlos aueaqtuatu oq& •osea aelno!la-ed qaea aoj paq!aasap se 'paanpoalu! aae lsaJalu! j o salqe!aen aql jo uo!lez!ieuaaou sno!ae A "saolenlae aql j o ssau)lo.tql aql pue tueoq aaAO[.Ilu130 aql j o qlguo I aql uoo~loq ~1/7 = a oIlea oql (!!) '. aolenlae aql jo ssamla!ql aql pue aanlanals aql j o ssau)D!ql aql uaatalaq "l/*l = ,L o!lea oql (.0 '. aJnlanals oMss-ed oql j o pue Ju3d saolenlae oql jo luotuo~uea.le oql oq.lJosop ol aopao I1.I tlosoq3 0.I13 SaOlOtrdeaed OAM puo s!ql o j+ "molsds oA.tldep'e aql JO ,Qlatl.lO0~ oql j o sttuol u! ~u!~ollOJ aql u! passnas!p aae uo!laas sno~Aaad aql u! pau!elqo suoIlnIos laexa aq& NOISSflDSI(I 'g
-uo!mlos Jop.lo lsalj oql slli.iol lsJg sl.t Ill su.n31uoo qo!qax 'uoIlnlos JapJo .toqg!q oql ql!ax luols!suoo s! llnsoa s!ql oseo gu!puaq oql u! o s l v (gg)
qv ~V r7 • s! ,,m -- = ~/oanleAan3 oql qo.~q~ moJ d
(~Tff)
ux'q I V
+vz _ = ( x > E7
zxqlv
~V = ~-~
(z'x)n
: oJ~ uo!13o~!p z 'x aql u.t luotua3~lds!p oql j o suotssoJdxa oq~, pu~ (¢;)
~V~ ~7
q l v - - -
0 ~
170M ~
~
E°dl4
~OM -~ IOM
qiv~V __-~7 0 =
t'Irt =
~[ n
gln =
lIn
: pou[elqo aae tuolqoad oql j o sluelsuoa lq~[o oql £IIeUId 0=(d
"P0at~+l £0AI ~+ ~0~1") D I + ( d l~I n+zl £I n+l ~I n+ I tn)t D ]
(z2)
q + g z " / = (zffln~+leinE+Zln)~v)~ l = x 0 = t o,~) 0 = t i n ~)
0=x
!zuapneD 'd
1709~
Exact higher order solutions for a simple adaptive structure
3605
1,1 0
1,08 1,06 C0
1,04 B
.....,
• ...,....
1,02
"'± •
-4
-3
o~._
-2
la B ~ ° I / a
,i
-1
0
1
2
3
4
Z
Fig. 2. Axial displacement (T = 5 ; r = 50 ; x = 0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
1,4 1,2 1 0,8 o
0,6
o
w#
0,4 "; •
I
I
t
-4
-3
-2
- - O f2-
,.j -1
O
°j-'
"".~)?2- ! . . . .
1
I 2
3
-0,4
Fig. 3. Axial strain (T = 5 ; r
=
50 ; x
=
0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
structure increases with respect to the E u l e r - B e r n o u l l i solution, as s h o w n also in Fig. 3. The extension o f this effect, which o n l y interest the edge o f the b e a m , d e p e n d s u p o n T ; the area interested to this effects increases with T. T h e pure b e n d i n g case is considered in Fig. 6 where the tip deflection o f the b e a m for several ratios r is s h o w n as a f u n c t i o n o f T. F r o m the picture it is possible to n o t e that significant differences with respect to the E u l e r - B e r n o u l l i m o d e l increase with T a n d a s s u m e larger values for low values o f r. I n Figs 7 a n d 8 the axial strain ( n o r m a l i z e d with respect to A1b) a n d the b e n d i n g stress ( n o r m a l i z e d with respect to E J[(1--•2)Alb]) are illustrated for r = 50 a n d T = 5 for several values o f x . I n a n a l o g y with the m e m b r a n e case the r e m o v a l o f plane section hypothesis is evident only in p r o x i m i t y o f the edge. F i g u r e 9 illustrates the i n d u c e d c u r v a t u r e ( n o r m a l i z e d with respect to 2Alb/ts) m e a s u r e d at x --- 0.95L for r = 50, as a f u n c t i o n o f T. Also in this case the differences b e t w e e n the
3606
P. Gaudenzi
0,6 "" f..o~
m -i
0,4 r
s I
I
I
'-I
I
0
-1 " ' ~ . . 1 ..-''1 -0,2 ~ "
-2
•
3,"
2
s
-0,4 -0,6 ..0,8
Fig, 4. Axial stress (T = 5 ; r = 50 ; x = 0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
.i
J#
;t o## m
• ~,w#~ f.
1 0 0
I
I
I
I
I
I
0,2
0,4
0,6
0,8
1
1,2
~L
Fig. 5. Axial strain (z = hi2 ; r = 50 ; T = 5 solid line ; T = 15 dashed line ; T = 30 dotted line).
E u l e r - B e r n o u l l i s o l u t i o n a n d the p r e s e n t one are i m p o r t a n t o n l y as T b e c o m e s large. T h e " P i n - f o r c e " results (1) are also s h o w n for c o m p a r i s o n . I n Fig. 10 the c u r v a t u r e is n o r m a l i z e d w i t h respect to A~b. E u l e r - B e r n o u l l i (solid line) a n d high o r d e r s o l u t i o n s h o w a r e l e v a n t difference o n l y as T increases as b e t t e r i l l u s t r a t e d in Fig. 11 w h e r e the c u r v a t u r e is p l o t t e d s t a r t i n g f r o m T = 5. F o r low values o f T the higher o r d e r a n d the E u l e r - B e r n o u l l i m o d e l s give the s a m e results. F r o m Fig. 12, in w h i c h the c u r v a t u r e is n o r m a l i z e d w i t h respect to the c o r r e s p o n d i n g E u l e r - B e r n o u U i solution, the e x t e n s i o n o f the edge effect for different values o f T c a n be e x a m i n e d . T h e larger is T the m o r e e x t e n d e d is the p o r t i o n o f the b e a m interested b y the effect. A s r a n d L are fixed (in this case L = 50, r = 50) the effect o f the increase o f T c a n be i n t e r p r e t e d as a shear d e f o r m a t i o n effect, r e p r e s e n t e d b y h i g h e r o r d e r t e r m s o f the p o w e r e x p a n s i o n o f d i s p l a c e m e n t c o m p o n e n t s in t e r m s o f z.
Exact higher order solutions for a simple adaptive structure
3607
1,35 1,3 1,25
t
1,2 1,15 1,1
1,05 1m 0
•_--..--..-'+-..-.'7."..-. . . . . 10
I" " r " " . . . . . . . 15
] 20
T Fig. 6. Normalized deflection (x = L; r = 10 solid line; r = 25 dashed line; r = 30 dotted line).
it
0,5 [ I
-4
-3 eI
J e w
.
''
*'T
. . . . .
iI
.rA
.°:2 -0,
1
2
3
4
15
Fig. 7. Axial strain (T = 5 ; r = 50 ; x = 0.9L solid line ; x = 0.95L dashed line ; x = L dotted line).
6. CONCLUSIONS T h e p r o p o s e d m o d e l , a l t h o u g h very simple f r o m the p o i n t o f view o f the f o r m u l a t i o n , h a s p e r m i t t e d one to e v a l u a t e the i m p o r t a n c e o f h i g h e r o r d e r effects in the m o d e l i n g o f the i n t e r a c t i o n between i n d u c e d strain a c t u a t o r s a n d a simple b e a m structure. T h e closed f o r m s o l u t i o n o b t a i n e d for the m e m b r a n e a n d b e n d i n g cases has a l l o w e d o n e to identify the m a i n g e o m e t r i c a l a n d m a t e r i a l p a r a m e t e r s o f the p r o b l e m a n d their role in the r e s p o n s e o f the system. By e x a m i n i n g the results o b t a i n e d with this m o d e l , the r a n g e o f a p p l i c a b i l i t y o f the classical E u l e r - B e r n o u l l i m o d e l are b e t t e r assessed. M o r e o v e r the m o d e l p e r m i t s one to p r e d i c t the extension o f the a r e a interested b y the edge effect. T h e m e c h a n i c a l i n t e r a c t i o n b e t w e e n surface b o n d e d i n d u c e d strain a c t u a t o r s a n d passive s t r u c t u r e is e v a l u a t e d in m o r e detail t h a n with classical b e a m m o d e l with p o s s i b l e a d v a n t a g e s for the design process.
3608
P. G a u d e n z i
.:
o,s 0,4
.
:
-#
:"
0.2
,'"
,
: -4
":
~i
.0.6-1-
.,
F i g . 8. A x i a l s t r e s s ( T = 5 ; r = 50 ; x = 0 . 9 L s o l i d line ; x = 0 . 9 5 L d a s h e d line ; x = L d o t t e d line).
1,~ 1 "b
0,8 "
~
0,6 0,4.
0,20
I
I
I
I
5
10
15
20
T
F i g . 9. N o r m a l i z e d
c u r v a t u r e Kt,/2A~b (r = 50 ; x = 0 . 9 5 L ; p i n - f o r c e : d o t t e d line ; E u l e r - B e r n o u l l i s o l i d line ; h i g h e r o r d e r : d a s h e d line).
:
1,5
1,2 0,9 0,6 0,3 0 0
F i g . 10. N o r m a l i z e d
I
I
I
I
I
I
3
6
9
12
15
18
c u r v a t u r e K/Ajb (r = 50 ; x = 0 . 9 5 L ; E u l e r - B e r n o u l l i d o t t e d line).
: s o l i d line ; h i g h e r o r d e r :
Exact higher order solutions for a simple adaptive structure
3609
0,3
°,,t
~0,2
0.1 ]
-,
0 5
I
I
I
I
I
8
11
14
17
20
Fig. 11. Normalized curvature K/Alb (r = 50 ; x = 0.95L ; Euler-Bernoulli : solid line ; higher order: dotted line).
3,5 3
;I "1
2,5. "
,r,=
;i
2
=## ,~
°°° ,IJ
1,5 1 0,5 0 0
~2
~4
~6
0,8
1
x/I
Fig. 12. Normalized curvature K/Keb (r = 50; T = 5 solid line; T = 15 dashed line; T = 30 dotted line).
Acknowledgement--The financial support of the research project "Controllo delle vibrazioni di semplici strutture intelligenti" (MURST- Italian Ministry for University and Research--Rif. 410/1995) is gratefully acknowledged.
REFERENCES Barboni, R., Gaudenzi, P. and Strambi, G. (1995) On the modeling of actuator-structure interaction in intelligent structures. Intelligent Materials and Systems, Techna, pp. 291-298. Bisegna, P. and Maceri, F. (1996) An exact three-dimensional solution for simply supported rectangular piezoelectric plates. A S M E Journal of Applied Mechanics 63, 628~638. Crawley, E. F. (1994) Intelligent structures for aerospace: a technology overview and assessment. AIAA Journal 32(8), 1689-1699. Crawley, E. F. and de Louis, J. (1987) Use of piezoelectric actuators as element of intelligent structures. A1AA Journal 25(10), 1373-1385. Denoyer, K. K. and Kwak, M. K. (1996) Dynamics modeling and vibration suppression of a slewing structure utilizing piezoelectric sensors and actuators. Journal of Soundand Vibration 189(1), 13-31. Gaudenzi, P. (1997) On the electromechanical response of active composite materials with piezoelectric inclusion. Computers and Structures 65(2), 157-168. Gaudenzi, P. and Bathe, K. J. (1995) An iterative finite element procedure for the analysis of piezoelectric continua. Journal of lntelligent Material System and Structures 6, 266-273.
3610
P. Gaudenzi
Gaudenzi, P., Barboni, R. and Mannini, A. (1995) A finite element evaluation of single-layer and multi-layer theories for the analysis of laminated plates. Composite Structures 30, 427~,40. Gaudenzi, P., Barboni, R., Carbonaro, R. and Accettella, S. (1997) Direct position and velocity feed-back control on an active beam with PZT sensors and actuators. International Forum on Aeroelasticity and Structural Dynamics, 17-20 June 1997, Vol. II, pp. 285-292 Rome, CEAS. Kim, J., Varadan, V. V. and Varadan, V. K. (1997) Finite element modeling of structures including piezoelectric active devices. International Journal for Numerical Methods in En#ineering 40, 817-832. Ray, M. C., Rao, K. M. and Samanta, B. (1993) Exact solution for static analysis of intelligent structures under cylindrical bending. Computers and Structures 47(6), 1031-1042. Reddy, J. N. (1987) A generalization of two-dimensional theories of laminated composite plates. Common Appl. Numer. Meth. 3, 173-181. Saravanos, D. A. and Heyliger, P. R. (1995) Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. Journal of lntelli#ent Material System and Structures 6, 350-363. Saravanos, D. A., Heyliger, P. R. and Hopkins, D. A. (1997) Layerwise mechanics and finite element for the dynamics analysis of piezoelectric composite plates. International Journal of Solids and Structures 34, 359-378.