- Email: [email protected]
Precise deflection analysis of beams with piezoelectric patches
Composite Structures 60 (2003) 135–143 www.elsevier.com/locate/compstruct
Precise deflection analysis of beams with piezoelectric patches Osama J. Aldraihem *, Ahmed A. Khdeir Department of Mechanical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
Abstract Precise deflection models of beams with n pairs of piezoelectric patches are developed analytically. To formulate the models, the first-order and higher-order beam theories are used. The analytical solutions are obtained with the aid of the state-space approach and Jordan canonical form. A case study is presented to evaluate the performance of the authorsÕ previously reported models. Through a demonstrative example, a comparative study of the first-order and higher-order beams with two piezoelectric pairs is attained. It is shown that the first-order beam lacks that ability to accurately predict the beam behavior in the region of the patches when compared with the results of the higher-order beam. Further applications of the solutions are presented by investigating the effects of patch lengths and locations on the deflected shape of beams with two piezoelectric pairs. For clamped–hinged and clamped–clamped (C–C) beams, the deflected shape switches from negative to positive when the piezoelectric pairs are moved toward the supports. The results of C–C beams show that increasing the patch lengths does not necessarily increase the deflection. The presented solutions can be used in the design process to obtain detailed deformation information of beams with various boundary conditions. Moreover, the presented analysis can be readily used to perform precise shape control of beams with n pairs of piezoelectric patches. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction The development of piezoelectric materials has constituted a revolution in sensing and actuation applications in recent years. The rapid response and high resolution, along with other properties such as large bandwidth and little power consumption, make piezoelectric materials increasingly popular as potential candidates for sensors and actuators. Piezoelectric patches are being increasingly utilized in many structures including aerospace applications, sporting goods, and MEMS applications. For instance, piezoelectric patches are incorporated in flexible structures to provide precision position control, to reject noise and vibration and to supply linear motion. When the piezoelectric patches are employed in precise position control, precise deflection analysis becomes necessary to predict the structure response. A rigorous analytical model can give precise information of the deformation. For example, the model can be used to obtain detailed deflection, slope and curvature change of a beam with piezoelectric patches. Unfortunately, the *
Corresponding author. Tel.: +966-1-467-6671; fax: +966-1-4676652. E-mail address: [email protected] (O.J. Aldraihem).
availability of analytical solutions to 1-D beam structures with piezoelectric patches is limited to Euler–Bernoulli and first-order (Timoshenko) beams. Crawley and Anderson [1] have developed analytical models of beams with piezoelectric actuators. The models illustrate the mechanics of Euler-Bernoulli beams with surface mounted and embedded actuators. The analytical results have been verified by carrying out static experiments. Chandrashekhara and Varadarajan [2] have presented a finite element model of a composite beam using a higher-order shear deformation theory. Piezoelectric elements have been used to produce a desired deflection in beams with clamped–free (C–F), clamped–clamped (C–C) and simply supported ends. Aldraihem et al. [3] have developed a laminated beam model using two theories; namely, Euler–Bernoulli beam theory and first-order beam theory (FOBT). Piezoelectric layers have been used to control the vibration in a cantilever beam. Abramovich [4] has presented analytical formulation and closed form solutions of composite beams with piezoelectric actuators. The beam model was based on the first-order shear theory. Shape control for beams with various boundary conditions have been demonstrated. Yang and Ngoi [5] have analytically addressed the shape control of Euler–Bernoulli beams. Ang et al. [6] have developed analytical solutions
0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0263-8223(02)00317-3
136
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
for the static deflections of beams. They have adopted the first-order shear deformation theory. The present study is a continuation of the authorsÕ recent work [7], in which the deflections of first-order and higher-order beams with one pair of piezoelectric patches were obtained. Although the presented solutions in investigation [7] were analytical, they were approximate in the sense of material properties. It was assumed that the beam stiffness is uniform and constant throughout the beam length. The aim of the present study is rigorously develop precisely analytical models and to present exact solutions of beams with n pairs of piezoelectric patches. The first-order and higher-order beam theories will be applied in the formulation. The accuracy of the approximated solutions given in [7] is evaluated by a comparison with the present exact solutions for beams with various boundary conditions. The deflections created by two piezoelectric patches will be presented for beams with various boundary conditions. The effect of patch length and location on the deflected shapes of the beam is investigated. The results of the proposed theories will be compared to demonstrate the effectiveness of each theory in predicting the beam deflection.
external mechanical loads. The material properties of the structure can be isotropic and specially orthotropic. Using the principle of stationary potential energy, the governing equations of higher-order beam theory (HOBT) with piezoelectric patches are derived as: 0
0
ðMx Þ n1 ðPx Þ Qx þ n2 Sx ¼ 0 00
0
0
n1 ðPx Þ þ ðQx Þ n2 ðSx Þ ¼ 0
ð1Þ
with the following associated boundary conditions at x ¼ 0, L Essential B:C: Natural B:C: / Mx n1 Px w0 n1 Px 0 w n1 ðPx Þ þ Qx n2 Sx
ð2Þ
where n1 ¼ 4=3h2 , n2 ¼ 3n1 and h is the total thickness of the beam structure. A prime on a quantity denotes ordinary differentiation with respect to x. The moment and force resultants in terms of displacement quantities are given by: Mx ¼ D11 /0 n1 F11 ð/0 þ w00 Þ Mxp Px ¼ F11 /0 n1 H11 ð/0 þ w00 Þ Pxp Qx ¼ ðA55 n2 D55 Þð/ þ w0 Þ
ð3Þ
Sx ¼ ðD55 n2 F55 Þð/ þ w0 Þ 2. Analytical formulation Consider the structure shown in Fig. 1. The structure is composed of a host beam and n pairs of piezoelectric patches perfectly bonded on the beam faces. Each pair of piezoelectric patches can possess its own material and geometrical properties. The piezoelectric patches are positioned with identical poling direction (z direction) and are excited by the same voltage. The structure is symmetric about its mid-plane and is not subjected to
where w denotes the transverse displacement of the beam mid-plane and / denotes the rotation of normal to the x-axis about the y-axis. The beam stiffnesses are defined by: Z h=2 e 11 dz ðD11 ; F11 ; H11 Þ ¼ b ðz2 ; z4 ; z6 Þ Q h=2 ð4Þ Z h=2 2 4 e ð1; z ; z Þ Q 55 dz ðA55 ; D55 ; F55 Þ ¼ b h=2
Fig. 1. The geometry of a beam with n pairs of piezoelectric patches.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
The piezoelectric stress resultants are expressed as Z h=2 e 11 E3 d31 dz ðMxp ; Pxp Þ ¼ b ðz; z3 Þ Q
2
ð5Þ
h=2
e½J x
where e 11 Q
Q2 ¼ Q11 12 ; Q22
e 55 ¼ Q55 Q
ð6Þ
and ci3 cj3 i; j ¼ 1; 2; 4; 5; 6 c33 c1i c13 i ¼ 2; 3 Qi3 ¼ ci3 c11
Qij ¼ cij
ð7Þ
where cij are the components of the stiffness matrix, d31 is the piezoelectric coefficient, E3 is the electric field applied across the thickness of a patch and b is the beam width. The state space concept in conjunction with the Jordan canonical form will be used to analyze the deflection of laminated beams with n pairs of piezoelectric patches. Using this approach, one can obtain the displacement quantities and their derivatives as:
2
0 6 c1 6 60 ½A ¼ 6 60 6 40 0
1 0 0 0 0 c3
0 0 0 0 0 0
0 c1 1 0 0 0
0 0 0 1 0 c3
3 0 c2 7 7 07 7 07 7 15 0
and e½J x is a block diagonal defined as
x 1 0 0 0 0
1 2 x 2
x 1 0 0 0
1 3 x 6 1 2 x 2
x 1 0 0
0 0 0 0 ekx 0
0 0 0 0 0
3 7 7 7 7 7 7 5
ð10Þ
ekx
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ c1 þ c3 þ c2 c3
ð11Þ
and the load vector fF g is given by 9 8 0 > > > > > > > f1 ðMxp Þ0 þ f2 ðPxp Þ0 > > > > > = < 0 fF g ¼ 0 > > > > > > > > 0 > > > > : p 00 p 00 ; f4 ðMx Þ þ f5 ðPx Þ
ð12Þ
the constants c1 , c2 , c3 , f1 , f2 , f4 , f5 are defined in [8]. Substituting 12n continuity conditions and three boundary conditions at each end (x ¼ 0, x ¼ L) for the desired combinations of boundary conditions, one has to solve 12n þ 6 simultaneous equations to find the constants k1 ; k2 ; . . . ; k12nþ6 . A similar procedure can be
8 9 8 k1 > > > > > > > > > > > k2 > > > > > > > = < > > k3 > ½J x > ½Me > > k4 > > > > > > > > > > > > k > > 5 > > > > ; : > > k > 6 9 > 9 8 8 > > / > k12i5 > > > > > > > > > > > > > > 0 > > > / > k12i4 > > > > > > > > > > > = < = < < Rx w k12i3 1 ½J x ¼ þ ½Me½J x e½J g ½M fF ðgÞg dg ½Me 0 w k > > > > 12i2 > > > > > > > > > > > > > > > k12i1 > w00 > > > > > > > > > ; > ; : 000 > : > > w k 12i 9 > 8 > > > k12iþ1 > > > > > > > > > > > k12iþ2 > > > > > > > = < > > k 12iþ3 > ½J x > ½Me > > k12iþ4 > > > > > > > > > > > k12iþ5 > > > > > > : ; : k12iþ6 where ½M is a modal matrix, which contains eigenvectors and generalized eigenvectors of the matrix ½A
1 60 6 60 ¼6 60 6 40 0
137
0 6 x < l1
l2i1 < x < l2i
i ¼ 1; . . . ; n
ð8Þ
l2i < x < l2iþ1
followed to analyze the deflection using the FOBT, the reader should refer to [7]. In the FOBT, one needs 8n continuity conditions and two boundary conditions at each end for the desired combinations of boundary conditions to determine the kÕs constants.
ð9Þ 3. Results and discussion The analytical solutions developed in the previous section will be used to investigate the deflection of beams
138
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
with C–F, hinged–hinged (H–H), clamped–hinged (C– H), and C–C boundary conditions. The following materials properties are used in the analysis: Aluminum [7]: E ¼ 70:3 GPa;
m ¼ 0:34
PZT5H [9]: c11 ¼ 126 GPa;
c12 ¼ 79:5 GPa;
c33 ¼ 117 GPa;
c44 ¼ 23 GPa;
c13 ¼ 84:1 GPa ;
d31 ¼ 274:8 ð1012 mV1 Þ The geometric configuration of the beam, as shown in Fig. 1, is assumed to be: L ¼ 0:1 m;
tb ¼ 0:016 m;
tP ¼ 0:001 m
The piezoelectric patches are assumed to have identical material properties and to have identical thicknesses. The voltage applied on each patch is 10 V. In the FOBT, a value of 5=6 was taken for the shear correction factor.
3.1. Comparison with approximate results The precise analytical solutions presented in the previous section are used to evaluate the accuracy of the approximate analytical solutions [7]. The deflection curves of beams with one pair of PZT5H patches are obtained. The effects of patch central locations and lengths on the results are examined. In study [7], the stiffnesses A55 and D11 are assumed to be invariant along the beam axis. To calculate these stiffnesses, the piezoelectric patches are assumed to cover the entire beam length. Fig. 2 shows the deflected shape of the first-order beams (FOBT). For C–F and H–H beams, the present exact results are identical to those obtained by the approximate analysis (see Figs. 3 and 5 in [7]). For C–H and C–C boundary conditions and for length a ¼ l2 l1 ¼ 0:01 m, changing the location of the piezoelectric pair gives slight difference between the
Fig. 2. The deflected shape of the FOBT for (a) C–H (a ¼ 0:01 m), (b) C–C (a ¼ 0:01 m), (c) C–H (d ¼ 0:05 m), (d) C–C (d ¼ 0:05 m) boundary conditions.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
deflection of the approximate and exact analysis. However, increasing the length of the patches makes a pronounced difference between the results of the approximate and the exact analysis. Since the beam of the approximate analysis appears stiffer than the actual beam, the approximate solution always underestimates the deflection. Figs. 3 and 4 shows the deflection curves of the higher-order beams (HOBT). When the beam ends are H–H and C–F, the approximate results agree quit well with those of the exact. As in the case of the FOBT, for C–H and C–C boundary conditions, changing the location of the piezoelectric pair gives small difference between the deflection of the approximate and exact analysis. However, increasing the length of the patches makes a significant difference between the results of the approximate and the exact analysis. It is observed that the deflection slope of the approximate solution is discontinuous at the patches edges.
139
3.2. Beams with two pairs of patches Beams with two pairs of piezoelectric patches are considered. First, the validity of the FOBT results is evaluated for beams with two pairs of identical lengths (a1 ¼ a2 ¼ 0:01 m) and located at d1 ¼ ðl1 þ l2 Þ=2 ¼ 0:025 m and d2 ¼ ðl3 þ l4 Þ=2 ¼ 0:075 m. Fig. 5 shows the deflected shape of the FOBT and the HOBT. When the axial location is far away from the positions of the piezoelectric pairs, the FOBT provides deflection results almost identical to those obtained by the HOBT. In the regions of the patches, the FOBT underestimates the deflection when compared with the results of the HOBT. These results indicate that the first-order beams are stiffer than the higher-order beams only in the patches regions. Since the FOBT lacks that ability to accurately predict the beam behavior at the region of the patches, the HOBT will be used next to investigate the effects of the
Fig. 3. The deflection curves of the HOBT for (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
140
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
Fig. 4. The deflection curves of the HOBT for (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
Fig. 5. The deflected shape of the FOBT and the HOBT for (a) C–F, (b) H–H, C–H, and C–C boundary conditions.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
141
Fig. 6. The effect of the pairsÕ central locations (d1 and d2 ) on the deflection of beams with (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
pair location and length on the deflection. Fig. 6 presents the effect of the pairÕs central locations (d1 and d2 ) on the deflection of beams with various boundary conditions. The length of the piezoelectric patches are fixed to a1 ¼ a2 ¼ 0:01 m. For C–F beams, altering the central locations has no influence on the tip displacements of the beams. For H–H beams, the deflection increases by moving the patches toward the center of the beams. It is interesting to note that the largest deflection is located on the patches regions. For C–H and C–C beams, the deflected shape switches from negative to positive when the piezoelectric pairs are moved toward the supports. Fig. 7 shows the effect of the pairs lengths on the deflected shape. The central location of pair 1 is fixed at d1 ¼ 0:025 m and of pair 2 is fixed at d2 ¼ 0:075 m. For C–F, H–H and C–H beams, the deflection increases when the patches lengths are increased. The maximum deflection is located at the patches regions, with the exception of the C–F beam. In C–H beams, the maxi-
mum deflection is always at the piezoelectric pair, which is closed to the hinged support. The results of C–C beams are interesting. With two piezoelectric pairs and for the considered parameters, the deflection of the beam midpoint is identically zero. Furthermore, increasing the patch lengths does not necessarily increase the deflection. In fact, piezoelectric pairs of lengths a1 ¼ a2 ¼ 0:02 m can generate deflection greater than that produced by pairs of lengths a1 ¼ a2 ¼ 0:04 m.
4. Conclusions Exact analytical solutions are presented for beams with n pairs of piezoelectric patches. The solutions rely on the first-order and higher-order beam theories. The precise analytical solutions presented in this work are used to evaluate the accuracy of the authorsÕ previously reported solutions [7]. For C–F and H–H beams, the
142
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
Fig. 7. The effect of the pairs lengths on the deflected shape for (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
present exact results are identical to those obtained by the approximate analysis [7]. Changing the location of the piezoelectric pair gives slight difference between the results of the approximate and exact analysis. However, increasing the length of the patches makes a pronounced difference between the results of the approximate and the exact analysis. The approximate analysis has one advantage over the exact analysis. The approximate analysis requires solving only six simultaneous equations irrespective of the number of patches and the type of boundary conditions. On the other hand, the exact analysis requires solving 12n þ 6 simultaneous equations to obtain the solution for a beam with n piezoelectric patches. Through a demonstrative example, a comparative study of the first-order and higher-order beams with two piezoelectric pairs is attained. It is observed that the first-order beam lacks that ability to accurately predict
the beam behavior in the region of the patches when compared with the results of the higher-order beam. Further applications of the solutions are presented by investigating the effects of patches lengths and locations on the deflected shape of beams with two piezoelectric pairs. For C–H and C–C beams, the deflected shape switches from negative to positive when the piezoelectric pairs are moved toward the supports. The results of C–C beams show that increasing the patch lengths does not necessarily increase the deflection. The presented solutions can be used in the design process to obtain detailed deformation information of beams with various boundary conditions. Moreover, the presented analysis can be readily used to perform precise shape control of beams with n pairs of piezoelectric patches. Finally, the present analysis can be used as benchmarks for approximate solutions obtained by the finite element method and other numerical techniques.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
References [1] Crawley EF, Anderson EH. Detailed models of piezoelectric actuation of beams. J Intell Mater Syst Struct 1990;1:4–25. [2] Chandrashekhara K, Varadarajan S. Adaptive shape control of composite beams with piezoelectric actuators. J Intell Mater Syst Struct 1997;8:112–24. [3] Aldraihem OJ, Wetherhold RC, Singh T. Distributed control of laminated beams: Timoshenko vs. Euler–Bernoulli theory. J Intell Mater Syst Struct 1997;8:149–57. [4] Abramovich H. Deflection control of laminated composite beams with piezoceramic layers––closed form solution. Compos Struct 1998;43:217–31.
143
[5] Yang S, Ngoi B. Shape control of beams by piezoelectric actuators. AIAA J 2000;38:2292–8. [6] Ang KK, Reddy JN, Wang CM. Displacement control of timoshenko beams via induced strain actuators. Smart Mater Struct 2000;9:981–4. [7] Khdeir AA, Aldraihem OJ. Deflection analysis of beams with extension and shear piezoelectric patches using discontinuity functions. Smart Mater Struct 2001;10:212–20. [8] Aldraihem OJ, Khdeir AA. Smart beams with extension and thickness-shear piezoelectric actuators. Smart Mater Struct 2000;9:1–9. [9] Electro Ceramic Division, Data for Designers, Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146.
Precise deflection analysis of beams with piezoelectric patches Osama J. Aldraihem *, Ahmed A. Khdeir Department of Mechanical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
Abstract Precise deflection models of beams with n pairs of piezoelectric patches are developed analytically. To formulate the models, the first-order and higher-order beam theories are used. The analytical solutions are obtained with the aid of the state-space approach and Jordan canonical form. A case study is presented to evaluate the performance of the authorsÕ previously reported models. Through a demonstrative example, a comparative study of the first-order and higher-order beams with two piezoelectric pairs is attained. It is shown that the first-order beam lacks that ability to accurately predict the beam behavior in the region of the patches when compared with the results of the higher-order beam. Further applications of the solutions are presented by investigating the effects of patch lengths and locations on the deflected shape of beams with two piezoelectric pairs. For clamped–hinged and clamped–clamped (C–C) beams, the deflected shape switches from negative to positive when the piezoelectric pairs are moved toward the supports. The results of C–C beams show that increasing the patch lengths does not necessarily increase the deflection. The presented solutions can be used in the design process to obtain detailed deformation information of beams with various boundary conditions. Moreover, the presented analysis can be readily used to perform precise shape control of beams with n pairs of piezoelectric patches. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction The development of piezoelectric materials has constituted a revolution in sensing and actuation applications in recent years. The rapid response and high resolution, along with other properties such as large bandwidth and little power consumption, make piezoelectric materials increasingly popular as potential candidates for sensors and actuators. Piezoelectric patches are being increasingly utilized in many structures including aerospace applications, sporting goods, and MEMS applications. For instance, piezoelectric patches are incorporated in flexible structures to provide precision position control, to reject noise and vibration and to supply linear motion. When the piezoelectric patches are employed in precise position control, precise deflection analysis becomes necessary to predict the structure response. A rigorous analytical model can give precise information of the deformation. For example, the model can be used to obtain detailed deflection, slope and curvature change of a beam with piezoelectric patches. Unfortunately, the *
Corresponding author. Tel.: +966-1-467-6671; fax: +966-1-4676652. E-mail address: [email protected] (O.J. Aldraihem).
availability of analytical solutions to 1-D beam structures with piezoelectric patches is limited to Euler–Bernoulli and first-order (Timoshenko) beams. Crawley and Anderson [1] have developed analytical models of beams with piezoelectric actuators. The models illustrate the mechanics of Euler-Bernoulli beams with surface mounted and embedded actuators. The analytical results have been verified by carrying out static experiments. Chandrashekhara and Varadarajan [2] have presented a finite element model of a composite beam using a higher-order shear deformation theory. Piezoelectric elements have been used to produce a desired deflection in beams with clamped–free (C–F), clamped–clamped (C–C) and simply supported ends. Aldraihem et al. [3] have developed a laminated beam model using two theories; namely, Euler–Bernoulli beam theory and first-order beam theory (FOBT). Piezoelectric layers have been used to control the vibration in a cantilever beam. Abramovich [4] has presented analytical formulation and closed form solutions of composite beams with piezoelectric actuators. The beam model was based on the first-order shear theory. Shape control for beams with various boundary conditions have been demonstrated. Yang and Ngoi [5] have analytically addressed the shape control of Euler–Bernoulli beams. Ang et al. [6] have developed analytical solutions
0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0263-8223(02)00317-3
136
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
for the static deflections of beams. They have adopted the first-order shear deformation theory. The present study is a continuation of the authorsÕ recent work [7], in which the deflections of first-order and higher-order beams with one pair of piezoelectric patches were obtained. Although the presented solutions in investigation [7] were analytical, they were approximate in the sense of material properties. It was assumed that the beam stiffness is uniform and constant throughout the beam length. The aim of the present study is rigorously develop precisely analytical models and to present exact solutions of beams with n pairs of piezoelectric patches. The first-order and higher-order beam theories will be applied in the formulation. The accuracy of the approximated solutions given in [7] is evaluated by a comparison with the present exact solutions for beams with various boundary conditions. The deflections created by two piezoelectric patches will be presented for beams with various boundary conditions. The effect of patch length and location on the deflected shapes of the beam is investigated. The results of the proposed theories will be compared to demonstrate the effectiveness of each theory in predicting the beam deflection.
external mechanical loads. The material properties of the structure can be isotropic and specially orthotropic. Using the principle of stationary potential energy, the governing equations of higher-order beam theory (HOBT) with piezoelectric patches are derived as: 0
0
ðMx Þ n1 ðPx Þ Qx þ n2 Sx ¼ 0 00
0
0
n1 ðPx Þ þ ðQx Þ n2 ðSx Þ ¼ 0
ð1Þ
with the following associated boundary conditions at x ¼ 0, L Essential B:C: Natural B:C: / Mx n1 Px w0 n1 Px 0 w n1 ðPx Þ þ Qx n2 Sx
ð2Þ
where n1 ¼ 4=3h2 , n2 ¼ 3n1 and h is the total thickness of the beam structure. A prime on a quantity denotes ordinary differentiation with respect to x. The moment and force resultants in terms of displacement quantities are given by: Mx ¼ D11 /0 n1 F11 ð/0 þ w00 Þ Mxp Px ¼ F11 /0 n1 H11 ð/0 þ w00 Þ Pxp Qx ¼ ðA55 n2 D55 Þð/ þ w0 Þ
ð3Þ
Sx ¼ ðD55 n2 F55 Þð/ þ w0 Þ 2. Analytical formulation Consider the structure shown in Fig. 1. The structure is composed of a host beam and n pairs of piezoelectric patches perfectly bonded on the beam faces. Each pair of piezoelectric patches can possess its own material and geometrical properties. The piezoelectric patches are positioned with identical poling direction (z direction) and are excited by the same voltage. The structure is symmetric about its mid-plane and is not subjected to
where w denotes the transverse displacement of the beam mid-plane and / denotes the rotation of normal to the x-axis about the y-axis. The beam stiffnesses are defined by: Z h=2 e 11 dz ðD11 ; F11 ; H11 Þ ¼ b ðz2 ; z4 ; z6 Þ Q h=2 ð4Þ Z h=2 2 4 e ð1; z ; z Þ Q 55 dz ðA55 ; D55 ; F55 Þ ¼ b h=2
Fig. 1. The geometry of a beam with n pairs of piezoelectric patches.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
The piezoelectric stress resultants are expressed as Z h=2 e 11 E3 d31 dz ðMxp ; Pxp Þ ¼ b ðz; z3 Þ Q
2
ð5Þ
h=2
e½J x
where e 11 Q
Q2 ¼ Q11 12 ; Q22
e 55 ¼ Q55 Q
ð6Þ
and ci3 cj3 i; j ¼ 1; 2; 4; 5; 6 c33 c1i c13 i ¼ 2; 3 Qi3 ¼ ci3 c11
Qij ¼ cij
ð7Þ
where cij are the components of the stiffness matrix, d31 is the piezoelectric coefficient, E3 is the electric field applied across the thickness of a patch and b is the beam width. The state space concept in conjunction with the Jordan canonical form will be used to analyze the deflection of laminated beams with n pairs of piezoelectric patches. Using this approach, one can obtain the displacement quantities and their derivatives as:
2
0 6 c1 6 60 ½A ¼ 6 60 6 40 0
1 0 0 0 0 c3
0 0 0 0 0 0
0 c1 1 0 0 0
0 0 0 1 0 c3
3 0 c2 7 7 07 7 07 7 15 0
and e½J x is a block diagonal defined as
x 1 0 0 0 0
1 2 x 2
x 1 0 0 0
1 3 x 6 1 2 x 2
x 1 0 0
0 0 0 0 ekx 0
0 0 0 0 0
3 7 7 7 7 7 7 5
ð10Þ
ekx
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ c1 þ c3 þ c2 c3
ð11Þ
and the load vector fF g is given by 9 8 0 > > > > > > > f1 ðMxp Þ0 þ f2 ðPxp Þ0 > > > > > = < 0 fF g ¼ 0 > > > > > > > > 0 > > > > : p 00 p 00 ; f4 ðMx Þ þ f5 ðPx Þ
ð12Þ
the constants c1 , c2 , c3 , f1 , f2 , f4 , f5 are defined in [8]. Substituting 12n continuity conditions and three boundary conditions at each end (x ¼ 0, x ¼ L) for the desired combinations of boundary conditions, one has to solve 12n þ 6 simultaneous equations to find the constants k1 ; k2 ; . . . ; k12nþ6 . A similar procedure can be
8 9 8 k1 > > > > > > > > > > > k2 > > > > > > > = < > > k3 > ½J x > ½Me > > k4 > > > > > > > > > > > > k > > 5 > > > > ; : > > k > 6 9 > 9 8 8 > > / > k12i5 > > > > > > > > > > > > > > 0 > > > / > k12i4 > > > > > > > > > > > = < = < < Rx w k12i3 1 ½J x ¼ þ ½Me½J x e½J g ½M fF ðgÞg dg ½Me 0 w k > > > > 12i2 > > > > > > > > > > > > > > > k12i1 > w00 > > > > > > > > > ; > ; : 000 > : > > w k 12i 9 > 8 > > > k12iþ1 > > > > > > > > > > > k12iþ2 > > > > > > > = < > > k 12iþ3 > ½J x > ½Me > > k12iþ4 > > > > > > > > > > > k12iþ5 > > > > > > : ; : k12iþ6 where ½M is a modal matrix, which contains eigenvectors and generalized eigenvectors of the matrix ½A
1 60 6 60 ¼6 60 6 40 0
137
0 6 x < l1
l2i1 < x < l2i
i ¼ 1; . . . ; n
ð8Þ
l2i < x < l2iþ1
followed to analyze the deflection using the FOBT, the reader should refer to [7]. In the FOBT, one needs 8n continuity conditions and two boundary conditions at each end for the desired combinations of boundary conditions to determine the kÕs constants.
ð9Þ 3. Results and discussion The analytical solutions developed in the previous section will be used to investigate the deflection of beams
138
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
with C–F, hinged–hinged (H–H), clamped–hinged (C– H), and C–C boundary conditions. The following materials properties are used in the analysis: Aluminum [7]: E ¼ 70:3 GPa;
m ¼ 0:34
PZT5H [9]: c11 ¼ 126 GPa;
c12 ¼ 79:5 GPa;
c33 ¼ 117 GPa;
c44 ¼ 23 GPa;
c13 ¼ 84:1 GPa ;
d31 ¼ 274:8 ð1012 mV1 Þ The geometric configuration of the beam, as shown in Fig. 1, is assumed to be: L ¼ 0:1 m;
tb ¼ 0:016 m;
tP ¼ 0:001 m
The piezoelectric patches are assumed to have identical material properties and to have identical thicknesses. The voltage applied on each patch is 10 V. In the FOBT, a value of 5=6 was taken for the shear correction factor.
3.1. Comparison with approximate results The precise analytical solutions presented in the previous section are used to evaluate the accuracy of the approximate analytical solutions [7]. The deflection curves of beams with one pair of PZT5H patches are obtained. The effects of patch central locations and lengths on the results are examined. In study [7], the stiffnesses A55 and D11 are assumed to be invariant along the beam axis. To calculate these stiffnesses, the piezoelectric patches are assumed to cover the entire beam length. Fig. 2 shows the deflected shape of the first-order beams (FOBT). For C–F and H–H beams, the present exact results are identical to those obtained by the approximate analysis (see Figs. 3 and 5 in [7]). For C–H and C–C boundary conditions and for length a ¼ l2 l1 ¼ 0:01 m, changing the location of the piezoelectric pair gives slight difference between the
Fig. 2. The deflected shape of the FOBT for (a) C–H (a ¼ 0:01 m), (b) C–C (a ¼ 0:01 m), (c) C–H (d ¼ 0:05 m), (d) C–C (d ¼ 0:05 m) boundary conditions.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
deflection of the approximate and exact analysis. However, increasing the length of the patches makes a pronounced difference between the results of the approximate and the exact analysis. Since the beam of the approximate analysis appears stiffer than the actual beam, the approximate solution always underestimates the deflection. Figs. 3 and 4 shows the deflection curves of the higher-order beams (HOBT). When the beam ends are H–H and C–F, the approximate results agree quit well with those of the exact. As in the case of the FOBT, for C–H and C–C boundary conditions, changing the location of the piezoelectric pair gives small difference between the deflection of the approximate and exact analysis. However, increasing the length of the patches makes a significant difference between the results of the approximate and the exact analysis. It is observed that the deflection slope of the approximate solution is discontinuous at the patches edges.
139
3.2. Beams with two pairs of patches Beams with two pairs of piezoelectric patches are considered. First, the validity of the FOBT results is evaluated for beams with two pairs of identical lengths (a1 ¼ a2 ¼ 0:01 m) and located at d1 ¼ ðl1 þ l2 Þ=2 ¼ 0:025 m and d2 ¼ ðl3 þ l4 Þ=2 ¼ 0:075 m. Fig. 5 shows the deflected shape of the FOBT and the HOBT. When the axial location is far away from the positions of the piezoelectric pairs, the FOBT provides deflection results almost identical to those obtained by the HOBT. In the regions of the patches, the FOBT underestimates the deflection when compared with the results of the HOBT. These results indicate that the first-order beams are stiffer than the higher-order beams only in the patches regions. Since the FOBT lacks that ability to accurately predict the beam behavior at the region of the patches, the HOBT will be used next to investigate the effects of the
Fig. 3. The deflection curves of the HOBT for (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
140
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
Fig. 4. The deflection curves of the HOBT for (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
Fig. 5. The deflected shape of the FOBT and the HOBT for (a) C–F, (b) H–H, C–H, and C–C boundary conditions.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
141
Fig. 6. The effect of the pairsÕ central locations (d1 and d2 ) on the deflection of beams with (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
pair location and length on the deflection. Fig. 6 presents the effect of the pairÕs central locations (d1 and d2 ) on the deflection of beams with various boundary conditions. The length of the piezoelectric patches are fixed to a1 ¼ a2 ¼ 0:01 m. For C–F beams, altering the central locations has no influence on the tip displacements of the beams. For H–H beams, the deflection increases by moving the patches toward the center of the beams. It is interesting to note that the largest deflection is located on the patches regions. For C–H and C–C beams, the deflected shape switches from negative to positive when the piezoelectric pairs are moved toward the supports. Fig. 7 shows the effect of the pairs lengths on the deflected shape. The central location of pair 1 is fixed at d1 ¼ 0:025 m and of pair 2 is fixed at d2 ¼ 0:075 m. For C–F, H–H and C–H beams, the deflection increases when the patches lengths are increased. The maximum deflection is located at the patches regions, with the exception of the C–F beam. In C–H beams, the maxi-
mum deflection is always at the piezoelectric pair, which is closed to the hinged support. The results of C–C beams are interesting. With two piezoelectric pairs and for the considered parameters, the deflection of the beam midpoint is identically zero. Furthermore, increasing the patch lengths does not necessarily increase the deflection. In fact, piezoelectric pairs of lengths a1 ¼ a2 ¼ 0:02 m can generate deflection greater than that produced by pairs of lengths a1 ¼ a2 ¼ 0:04 m.
4. Conclusions Exact analytical solutions are presented for beams with n pairs of piezoelectric patches. The solutions rely on the first-order and higher-order beam theories. The precise analytical solutions presented in this work are used to evaluate the accuracy of the authorsÕ previously reported solutions [7]. For C–F and H–H beams, the
142
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
Fig. 7. The effect of the pairs lengths on the deflected shape for (a) C–F, (b) H–H, (c) C–H, (d) C–C boundary conditions.
present exact results are identical to those obtained by the approximate analysis [7]. Changing the location of the piezoelectric pair gives slight difference between the results of the approximate and exact analysis. However, increasing the length of the patches makes a pronounced difference between the results of the approximate and the exact analysis. The approximate analysis has one advantage over the exact analysis. The approximate analysis requires solving only six simultaneous equations irrespective of the number of patches and the type of boundary conditions. On the other hand, the exact analysis requires solving 12n þ 6 simultaneous equations to obtain the solution for a beam with n piezoelectric patches. Through a demonstrative example, a comparative study of the first-order and higher-order beams with two piezoelectric pairs is attained. It is observed that the first-order beam lacks that ability to accurately predict
the beam behavior in the region of the patches when compared with the results of the higher-order beam. Further applications of the solutions are presented by investigating the effects of patches lengths and locations on the deflected shape of beams with two piezoelectric pairs. For C–H and C–C beams, the deflected shape switches from negative to positive when the piezoelectric pairs are moved toward the supports. The results of C–C beams show that increasing the patch lengths does not necessarily increase the deflection. The presented solutions can be used in the design process to obtain detailed deformation information of beams with various boundary conditions. Moreover, the presented analysis can be readily used to perform precise shape control of beams with n pairs of piezoelectric patches. Finally, the present analysis can be used as benchmarks for approximate solutions obtained by the finite element method and other numerical techniques.
O.J. Aldraihem, A.A. Khdeir / Composite Structures 60 (2003) 135–143
References [1] Crawley EF, Anderson EH. Detailed models of piezoelectric actuation of beams. J Intell Mater Syst Struct 1990;1:4–25. [2] Chandrashekhara K, Varadarajan S. Adaptive shape control of composite beams with piezoelectric actuators. J Intell Mater Syst Struct 1997;8:112–24. [3] Aldraihem OJ, Wetherhold RC, Singh T. Distributed control of laminated beams: Timoshenko vs. Euler–Bernoulli theory. J Intell Mater Syst Struct 1997;8:149–57. [4] Abramovich H. Deflection control of laminated composite beams with piezoceramic layers––closed form solution. Compos Struct 1998;43:217–31.
143
[5] Yang S, Ngoi B. Shape control of beams by piezoelectric actuators. AIAA J 2000;38:2292–8. [6] Ang KK, Reddy JN, Wang CM. Displacement control of timoshenko beams via induced strain actuators. Smart Mater Struct 2000;9:981–4. [7] Khdeir AA, Aldraihem OJ. Deflection analysis of beams with extension and shear piezoelectric patches using discontinuity functions. Smart Mater Struct 2001;10:212–20. [8] Aldraihem OJ, Khdeir AA. Smart beams with extension and thickness-shear piezoelectric actuators. Smart Mater Struct 2000;9:1–9. [9] Electro Ceramic Division, Data for Designers, Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146.