Three-Dimensional Bending Analysis of

Functionally-Modified Bimorph PZT Actuator for cm-Scale 

Flapping Wing

by Jaret C. Riddick and Asha Hall

ARL-TR-5581 July 2011

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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5069

ARL-TR-5581 July 2011

Three-Dimensional Bending Analysis of

Functionally-Modified Bimorph PZT Actuator for cm-Scale 

Flapping Wing

Jaret C. Riddick

Vehicle Technology Directorate, ARL

Asha Hall

Motile Robotics, Inc.

Approved for public release; distribution is unlimited.
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4. TITLE AND SUBTITLE

Three-Dimensional Bending Analysis of Functionally-Modified Bimorph PZT 

Actuator for cm-Scale Flapping Wing

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6. AUTHOR(S)

Jaret C. Riddick and Asha Hall*

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U.S. Army Research Laboratory

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*Motile Robotics, Inc., Joppa, MD, 21085

14. ABSTRACT

Army combat operations have placed a high premium on reconnaissance missions for micro air vehicles (MAVs). An analysis of insect flight 

indicates that in addition to the bending excitation (flapping), simultaneous excitation of the twisting degree-of-freedom is required to 

manipulate the control surface adequately. By adding a layer of angled piezoelectric segments to a Pb(Zr,Ti)O3(also referred to as PZT) 

bimorph actuator, a bend-twist coupling may be introduced to the flexural response of the layered PZT, thereby creating a biaxial actuator 

capable of driving wing oscillation in flapping wing MAVs. The present study presents numerical solutions to governing equations for quasistatic three-dimensional bending of functionally–modified bimorph designs intended for active bend-twist actuation of cm-scale flapping wing 

devices. The results indicate a strong dependence of bimorph deflection on overall length. Further, the width and angle of orientation of the 

angled piezoelectric segments may be manipulated in order to increase or decrease the length effects on bimorph deflection. The relationships 

of geometry and orientation of the angled segments with bimorph flexural response are presented.

15. SUBJECT TERMS

piezoelectric, bimorph actuator, three-dimensional bending, functionally-modified, PZT

16. SECURITY CLASSIFICATION OF: 

17. LIMITATION

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 ABSTRACT

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19a. NAME OF RESPONSIBLE PERSON

Jaret Riddick

a. REPORT

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(410) 278-9831

Standard Form 298 (Rev. 8/98)

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iii

Contents

List of Figures iv

List of Tables iv

1. Introduction 1

2. Geometry and Nomenclature 3

3. Analysis 5

4. Results and Discussion 8

5. Conclusions 11

6. References 12

Distribution List 13
iv

List of Figures

Figure 1. Conventional bimorph actuator. ......................................................................................1

Figure 2. Flapping wing device. .....................................................................................................2

Figure 3. Functionally-modified bimorph.......................................................................................3

Figure 4. Coordinate system. ..........................................................................................................3

Figure 5. Representative element....................................................................................................4

Figure 6. Three representative element typical cross-sections. ......................................................4

Figure 7. Upper and lower bounds on angled segment orientation angle θ...................................9

Figure 8. Deflection results for Cases II, III, and IV. ...................................................................10

Figure 9. Effect of orientation angle θ on functionally-modified bimorph deflection for 

L/b = 1.3...................................................................................................................................11

List of Tables

Table 1. Representative element results for b = 6 mm exposed to P = 1.0 N concentrated 

load in negative z-direction (downward). ..................................................................................9
1

1. Introduction

Significant technical strides have been accomplished in designing cm-scale micro air vehicles 

(MAVs). One approach for accomplishing this mission is to develop a biologically inspired 

flapping wing insect that can maneuver into confined areas and possess hovering capabilities. 

Insect-like MAVs have great potential for limited duration electronic surveillance and sensorbased detection for both military and civilian missions (1). Some advantages of MAVs include 

portability, low noise, low altitude aerial surveillance, low radar cross section, low power 

consumption, and ease of generation of lift or thrust at low weight. MAVs are implemented as 

fixed, rotating, or flapping wing solutions. Fixed wing MAVs use the wing span, the angle of 

attack as a function of forward airspeed, and the surface area to gain lift and thrust; meanwhile,

flapping wing systems generate lift through the flapping wing beat frequency. A significant 

advantage of the flapping wing systems is lower bandwidth where the cyclic control inputs to 

maintain lift and stability operate at lower frequencies as compared with a rotor system (2). 

However, due to the size limitations of MAVs (typically, less than 15 cm in dimension and less 

than 20 g in mass) it is a challenge for the flapping wing vehicle to perform in adverse weather 

conditions (i.e., wind gust and precipitation). One tactic is to provide adaptability or shape 

control of the wing structure that can create effective aerodynamic forces (3).

Analysis of insect flight indicates that in addition to the bending excitation (flapping), 

simultaneous excitation of the twisting degree-of-freedom (pitching) is required to manipulate 

the control surface adequately (2). Traditionally, bimorph piezoelectric Pb(Zr,Ti)O3 (also 

referred to as PZT) actuators have been used in many applications to excite the bending degreeof-freedom. A common bimorph configuration, as shown in figure 1, consists of two thin 

ceramic plates bonded together and driven with opposing electrical fields. One plate expands 

while the other contracts, resulting in a bending deflection of the bimorph construction. As a 

result, a flapping motion is generated from operating at the fundamental resonance frequency of 

the system (4).

Figure 1. Conventional bimorph actuator.
2

In prior work, a flapping mechanism was developed that couples a 3 cm PZT-5H bimorph 

actuator oscillating at resonance frequency with flexural wing joints placed at the distal end of 

the bimorph (see figure 2). However, in the current approach bend-twist coupling arises from 

anisotropic material symmetry. In laminated or layered structures, bend-twist coupling is 

governed by the existence of at least one anisotropic layer not aligned with the primary plate 

axes (5). By adding to a bimorph PZT actuator a layer of off-axis PZT segments active, thereby 

producing a layered structure to be referred to as a functionally–modified bimorph, bend-twist 

coupling may be introduced to the flexural response of the layered PZT. Furthermore, by 

selectively charging off-axis layers in specific combinations with the bimorph, the response of 

the functionally–modified bimorph may be tailored yielding a biaxial actuator to actively control 

the flapping wing response. 

Figure 2. Flapping wing device.

Several studies have considered the use of piezoelectric actuators to drive the bending degree-offreedom of cantilevered beams for applications ranging from control surfaces of aerospace 

structures (6) to smart rotorcraft blades (7, 8). Piezoelectric cantilevered bimorph actuators for 

optical deflection and modulation in MEMS applications have been investigated by applying an 

Euler-Bernoulli beam analysis to clamped bender configurations (9). Studies have also been 

presented that use energy methods (10) and elasticity solutions (11) to develop quasi-static 

bending solutions that yielded parametric relationships between tip deflection, applied loads and 

moments, actuator geometrical dimensions and material properties. The present study develops 

a numerical solution to the three-dimensional bending analysis of a cantilevered functionallymodified bimorph to investigate the effects of angled segment geometry and orientation on the 

bimorph bending response.
3

2. Geometry and Nomenclature

A conventional PZT bimorph cantilever configuration functionalized by the addition of angled 

PZT segments shown in figure 3 was considered. The coordinate axis referred to as x-y-z is 

shown in figure 4, where the origin is fixed at the clamped end of the functionalized bimorph at 

the base of the bottom layer. For the purpose of analysis, the functionally-modified bimorph was 

idealized as a homogeneous base with homogeneous angle segments attached on top, consisting 

of transversely isotropic PZT-5H material (Young’s modulus, E = 65 GPa). The stainless steel 

shim depicted in figure 3 was ignored

Figure 3. Functionally-modified bimorph.

The distance from the origin to the midplane of the base along the z-axis was h. The total 

thickness of the base bimorph was 2h, and the thickness of the angled segments was h. The 

width b shown in figure 3 relates to the thickness h as follows: b = 0.1 h. The angle of the PZT 

segments was  as shown in figure 4. The width of each angled segment measured along the 

x-axis was d, as shown in figure 5. The overall length LT of the functionalized bimorph included 

five angled segments separated by distance s as shown in figure 4.

Figure 4. Coordinate system.
4

In order to develop section properties and geometrical relationships useful for design of PZT 

configurations, a representative element of the functionalized bimorph was defined to consist of 

a single angled segment affixed to a bimorph layer shown in figure 5.

Figure 5. Representative element.

The representative element length, L, shown in figure 5, was divided into three distinct regions, 

namely, sections 1, 2, and 3. Typical cross-sections for are shown in figure 6, where the shaded 

regions represent the angled segment. Using the described geometry and the points A, B, C, and 

D and points Q, R, S, and T shown in figure 5, functional relations for y-axis locations along the 

edges of the angle segments were derived, and are presented below. Section 1 (0 ≤ x < d), was 

bounded in y-z by the plane containing the origin O and the points A and C at x = 0 and the plane 

containing points Q and T at x = d. As x increased from 0 to d, the width of the angled segment 

increased from 0 to t with respect to the y-axis. The section 1 cross-section typified by the y-z

plane at A- A in figure 6 was characterized by the angled segment where the y-location of edge 

AQ was fixed at y = b/2 and interior edge AT was a function of x:



yAT (x) 

b

2



t

d

x

. (1)

Figure 6. Three representative element typical cross-sections.
5

Section 2 (d ≤ x ≤ L  d) was bounded in y-z by the plane containing points Q and T at x = d, and 

at x = L  d by the plane containing points R and S. As x increased from d to L  d, the width of 

the angled segment remained constant at t. The section 2 cross-section typified by the y-z plane 

B-B in figure 6 was characterized by the angled segment where the y-locations of both edges 

QR and TS were functions of x:



y

QR(x) 

t b

L2d

(xd)

b

2

(2)



yTS (x) 

b  t

L  2d

(L  d  x) 

b

2

(3)

Section 3 (L  d < x ≤ L) was bounded in y-z by the plane containing the points R and S at 

x = L  d, and the plane containing points B and D at x = L. As x increased from L  d to L, the 

width of the angled segment decreased from t to 0 with respect to the y-axis. The section 3 

cross-section typified by the y-z plane C- C in figure 6 was characterized by the angled 

segment where the y-location of interior edge RD is a function of x and edge SD is fixed at 

y = -b/2:



yR D(x) 

t

d

(L  x) 

b

2

. (4)

The following relationships defined axial and transverse width parameters d and t,

(5)

tan

2

t d         

. (6)

The geometry of the representative element prescribed constraints on the angled segment width 

parameters d and t—namely, d > 0 and t < b. Applying the constraints on the width to the 

relationships of equations 5 and 6 yielded the following bounds on , thus defining angled 

segment orientation,

cot (2 / ) tan ( / )

1 1

b L L b

 

  . (7)

3. Analysis

The results of traditional bending analyses have been used to relate bimorph loading and 

geometry. For the functionally-modified bimorph considered in the present analysis, the crosssection of the cantilevered bimorph was not symmetric in respect to the horizontal (x-y) plane. 

Therefore, a three-dimensional bending analysis was applied in order to relate the actuator 

loading and geometric properties to tip deflection (12). In order to gain a complete 



d  Lbtan
6

understanding of the complex three-dimensional analysis the piezoelectric effects will be not be 

considered here.

Beginning with the Euler-Bernoulli assumptions—(1) the cross-section is infinitely rigid in its 

own plane; (2) the cross-section of a beam remains plane after deformation; and (3) the crosssection remains normal to the deformed axis of the beam—it was assumed that no torsional loads 

are applied and that the beam bends without twisting. Finally making the assumption that the 

origin of the coordinate axis system was selected to align with the centroid of the beam as shown 

in figure 4, the following set of equations govern three-dimensional bending (12):

(8)

(9)

where, 



v 

(x) and 



w 

(x) represented displacements of the beam cross-section in the y- and zdirections, respectively. Distributed loads along the x-axis in the x-, y-, and z-directions were 

defined as px(x), py(x), and pz(x), respectively. Coordinates (ya, za) defined the location in the y-z

plane of the axial distributed load px(x). Distributed moments along the x-axis about the y- and zaxes were defined as qy(x) and qz(x), respectively. The boundary conditions for the fixed end of 

the cantilevered bimorph were given by

(10)

At the free end, the boundary conditions were as follows (with the shorthand notation in the 

parentheses) (12):

(11)

where Py and Pz represented applied shear at the tip in the respective y- and z-directions. Applied 

bending moments in the y- and z-directions were represented by Qy and Qz, respectively. In the 

shorthand notation (equation 11 in parentheses), M and V represented the internal moments and 

shears, respectively. The centroidal bending stiffness coefficients Hyy, Hzz, and Hyz were defined



v  w  0; dv dx dw dx 0;



Hzz

c

d

2

v 

d x2

 Hy z

c

d

2

w 

d x2

 Qz yaPx Mz Qz y  aPx 

Hy z

c

d

2

v 

d x2

 Hy y

c

d

2

w 

d x2

 Qy zaPx My Qy z  aPx 



d

d x

Hzz

c

d

2

v 

d x2

 Hy z

c

d

2

w 

d x2







 Py

 yapx q  z Vy Py 



d

d x

Hy z

c

d

2

v 

d x2

 Hy y

c

d

2

w 

d x2







 Pz

 zapx q  y  Vz Pz

d

2

dx 2 Hyz

c d

2

v 

dx 2

 Hyy

c d

2

w 

dx 2







 pz



d

dx

za px   qy 

d

2

dx 2 Hzz

c d

2

v 

dx 2

 Hyz

c d

2

w 

dx 2







 py 

d

dx

y a px  qz
7

(12)

where, Hyy, Hzz, and Hyz represent the bending stiffness coefficients with respect to an arbitrary 

coordinate axis:

(13)

where, E was the Young’s modulus and y and z refer to coordinates of an arbitrary point in the 

y-z plane. Note here that the bending stiffness coefficients are functions of beam cross-sectional 

area in the y-z plane. For the functionally-modified bimorph, as the bimorph geometry traverses 

the x-axis, the beam cross-section in y-z is dependent on the relationships for y as a function of x

developed in equations 1–4. Therefore, the coefficients of the ordinary differential governing 

equations of equations 8 and 9 and the boundary conditions in equation 11 are functions of x, and 

are considered to be variable coefficients. Generally, ordinary differential equations with 

variable coefficients must be solved numerically.

The sectional stiffness coefficients S, Sy, and Sz are defined as follows (note that the coefficient 

Sy is a function of x):

(14)

the centroid of a the beam cross-section (yc,zc) as shown in figure 4 is defined in terms of section 

stiffness coefficients

. (15)

Again, note that the centroidal location yc varies with x-location along the beam. In order to 

perform the three-dimensional bending analysis of the functionally-modified bimorph, the beam 

was considered in sections along its length (x-axis) by defining a representative element, as 

shown in figure 5. The governing equations applied to the sections of the representative element 

shown in figure 5, take on the following piece-wise form:



0  x  d :

d

2

d x2

Hzz

c d

2

v 1

d x2

 Hyz

c d

2

w 1

d x2







 0

d

2

d x2

Hyz

c d

2

v 1

d x2

 Hyy

c d

2

w 1

d x2







 0

(16)



d  x  L  d :

d

2

dx 2

Hzz

c d

2

v 2

dx 2

 Hyz

c d

2

w 2

dx 2

 

   

 

    0

d

2

dx 2

Hyz

c d

2

v 2

dx 2

 Hyy

c d

2

w 2

dx 2

 

   

 

    0

(17)



Hyy Hyy

c

 Szc

2

; Hz z Hz z

c

 Syc

2

; Hyz Hyz

c

 Syczc;



yc

Sy

S

zc

Sz

S

S  EdA;

A

 Sy  EydA;

A

 Sz  EzdA

A

 ;

Hyy  Ez2dA;

A

 Hzz  Ey 2

dA;

A

 Hyz  EyzdA

A



;
8



L  d  x  L :

d

2

dx 2

Hzz

c d

2

v 3

dx 2

 Hyz

c d

2

w 3

dx 2

 

   

 

    0

d

2

dx 2

Hyz

c d

2

v 3

dx 2

 Hyy

c d

2

w 3

dx 2

 

   

 

    0

(18)

where, the subscripts on (x) and (x) represent the sections of the representative element as 

depicted in figure 5. The boundary conditions for a concentrated load P applied to the free end 

of the cantilevered bimorph configuration assumed, using the shorthand notation from equation

11



x  0 : v 1 w 1 0; dv 1/d x  dw 1/ d x  0;

x  LT: M y3 0

M z3 0

Vy3 0

Vz3 P

.

(19)

Additionally, the following continuity conditions were applied to allow the analysis to span the 

sections of the piece-wise representative element, where the displacements and slopes at the 

junctions of the segments were defined as being equal:



x  d : v 1  v 2; w 1  w 2; dv 1 /d x dv 2 /d x; dw 1 /d x dw 2 /d x;

x  Ld : v 2  v 3; w 2  w 3; dv 2 /d x dv 3 /d x; dw 2 /d x dw 3 /d x.

(20)

In order, to find the deflection of the entire functionally-modified bimorph, continuity conditions 

were applied to a series of representative elements with spaces of traditional bimorph in-between 

as shown in figure 4. The result of this process was a system of 38 fourth order coupled variable 

coefficient ordinary differential equations (ODEs). The system of ODEs was solved numerically 

using the built-in symbol NDSolve in Mathematica* Version 8 (13) to find the tip displacement 

in terms of bimorph geometrical parameters. Results for the deflection of the cantilevered 

functionally-modified bimorph in respect to angled segment geometry and orientation are 

presented in the next section.

4. Results and Discussion

The angled segments of the functionally-modified bimorph formed an angle  with the clamped 

edge of the bimorph as shown in the inset in figure 7. Equations 5 and 6 provided the 

relationships between the angle  and geometric parameters that govern length L, angled 

segment widths b and t. Figure 7 shows the upper and lower bounds on  as defined by 

equations 5 and 6, respectively. The points labeled in figure 7 as I–VII correspond to the case 

numbers in table 1.

 

* Mathematica is a trademark of Wolfram Research, Inc.



v 



w 
9

Figure 7. Upper and lower bounds on angled segment orientation angle θ

Presented in table 1 are three-dimensional bending results for the bimorph configuration of 

figure 4, where b was fixed at 6 mm and exposed to an applied tip shear P = 1.0 N in the 

negative z (downward) direction. Cases I and II (labeled in figure 6) shared the same 

orientation for different ratios of length L and width b. Because b was fixed, decreasing L/b ratio 

indicated increasing L; therefore, the length of the representative element in Case II was greater. 

The maximum section displacement results occurred at the x = LT tip location. The results for 

Cases I and II predicted that maximum deflection increases as length L increases, indicating a 

strong dependence on overall length LT.

Table 1. Representative element results for b = 6 mm exposed to P = 1.0 N concentrated 

load in negative z-direction (downward).

Case L/b  d/L vmax (10–6 m) wmax (10–3 m)

I 0.9 34 0.35 0.355 0.0572

II 1.3 34 0.48 0.161 0.121

III 1.3 45 0.23 0.134 0.148

IV 1.3 51 0.050 0.997 0.134

V 1.9 45 0.47 0.643 0.374

VI 1.9 51 0.35 2.239 0.331

VII 1.9 60 0.085 2.667 0.322

Comparison of Cases III and V of table 1, also labeled in figure 7, further evidenced the 

dependence of the maximum deflection value on bimorph length. Cases III and V shared the 

same value of . The length of the Case V configuration was greater than Case III. The 

maximum deflection increased as length increased. Also, considering Cases IV and VI, where 

values were similar between the two cases. The length of Case VI was greater, and maximum 

deflection for Case VI was greater. The results in table 1 clearly establish that there is a strong 
10

dependence of maximum deflection on overall bimorph length. Namely, as length increases, 

maximum deflection increases.

Figure 8 presents the deflection curves for Cases II, III, and IV, where the  values are given by 

34, 45, and 51 respectively. Each case, as shown in table 1, had a L/b ratio of 1.3, implying an 

overall length LT = 28 mm. Here, as  increased, d/L decreased, indicating a decrease in the 

width of the angled segment. Intuitively, the increase in  would seem to imply an increase in 

the stiffening capability of the segment. However, as shown in figure 8, Case III at  = 45, 

d/L = 0.23, displayed the greatest deflection and was therefore the least stiff configuration, while 

for Cases II and IV, both configurations displayed greater stiffness, compared with Case III, as 

segment width increased and decreased, respectively.

Figure 8. Deflection results for Cases II, III, and IV.

Figure 9 shows the relationship of deflection and angled segment orientation  for L/b = 1.3. 

The data showed that as  increased between the bounds prescribed by the representative 

element geometry, the deflection reached a maximum at around  = 45 and decreased until it 

reached the upper bound on . As increased, the data in figure 9 show that the width parameter 

d/L indicated decreasing width d. The change in the width of the angled segment resulted in a 

change in the section bending stiffness coefficients Hyy, Hzz, and Hyz defined in equation 12. As 

increased to  = 45, the section bending stiffness coefficients decreased to a value that nearly 

coincided with the section properties for the traditional bimorph (without the angled segments). 

Therefore, due to the change in angled segment width brought about by the change in angle , 

the section properties influenced the bending response so that the resistance to bending in the 

bimorph decreased. Thus, the orientation and width of the angled segments interacted in such a 

way that the bending capacity of the traditional bimorph served as a lower bound on deflection. 
11

The increase in deflection after  = 45 may have been due to error typical to numerical 

solutions.

Figure 9. Effect of orientation angle θ on functionally-modified bimorph deflection for L/b = 1.3.

5. Conclusions 

A quasi-static three-dimensional bending analysis of a functionally-modified bimorph 

configuration has been presented. The analysis was conducted on an idealized configuration 

where angled PZT segments were assumed to be affixed to a base traditional bimorph 

configuration. The presence of the metal shim was neglected analytically. In general, bimorph 

deflection increased with increase in bimorph length. For fixed bimorph length LT, as 

increases, deflection increases, due to changes in the bimorph sectional stiffness properties. The 

prescribed geometry of the functionally-modified bimorph configuration provided constraints on 

the orientation angle of the PZT segments affixed to the bimorph surface. The orientation and 

width of the angled segments interacted in such a way that the bending properties of the 

traditional bimorph act as a lower bound on deflection. Future work will include numerical 

validation of the three-dimensional bending analysis via finite-element analysis and experimental 

validation, with the ultimate goal of developing a design tool for biaxial actuators for cm-scale 

flapping wing MAVs.
12

6. References

1. Mueller, T. J. Fixed and Flapping Wing for Micro Air Vehicles, AIAA Inc.: Reston, VA, 

2001, pp 1–10.

2. Wu, P.; Ifju, P. Flapping Wing Effectiveness, Efficiency and Aeroelasticity Relationships. 

48th AIAA Aerospace Sciences Meeting, Orlando, FL, January 4–7, 2010, (1017), pp 1–15. 

3. Lewitowicz, J.; Kowalczyk, G.; Sibilski, K.; Zurek, J. Modeling and Simulation of Flapping 

Wings Micro-aerial-vehicles Flight Dynamics. 26th International Congress of the 

Aeronautical Sciences (ICAS), Anchorage, AK, September 14–19, 2008; pp 1–24.

4. Hall, A.; Pekala, M. C. Micro-electro Mechanical Flapping Wing Technology for Micro Air 

Vehicles. IMAV 2010 Proceedings, Braunschweig, Germany, July 6–9, 2010; pp 1–6. 

5. Wetherhold, R. C.; Aldraheim, O. J. Bending and Twisting Vibration Control of Flexible 

Structures Using Piezoelectric Materials. Shock and Vib. Dig. 2001, 33 (3), 187–197.

6. Crawley, E. F.; de Luis, J. Use of Piezoelectric Actuators as Elements of Intelligent 

Structures. AIAA J. 1987, 25 (10) 1373–1385.

7. Aldraheim, O. J.; Wetherhold, R. C. Mechanics and Control of Coupled Bending and 

Twisting Vibration of Laminated Beams. Smart Mtls. and Struct. 1997, 6, 123–133.

8. Park, C.; Walz, C.; Chopra, I. Bending and Torsion Models of Beams with Induced-Strain 

Actuators. Smart Mtls. and Struct. 1996, 5, 98–113.

9. Ballato, A.; Kim, Y. Displacements and Rotations of Practical Vibrational Modes of 

Piezoelectric Bimorph Cantilever Beams. Proceedings of First IEEE International 

Conference on Sensors, Orlando, FL, June 12–14, 2002, 2, pp 1294–1297.

10. Smits, J. G.; Dalke, S. I.; Cooney, T. K. The Constituent Equations of Piezoelectric 

Bimorphs. Sens. and Act. A 1991, 28, 41–61.

11. Wang, Q.-M.; Cross, L. E. Performance Analysis of Piezoelectric Cantilever Bending 

Actuators. Ferroelec. 1998, 215 (1), 187–213.

12. Bauchau, O. A.; Craig, J. I. Structural Analysis: With Applications to Aerospace Structures;

Springer: Dordrecht, Heidelberg, London & New York, 2009, pp 171–202.

13. Wolfram, S. The Mathematica Book; Wolfram Media/Cambridge University Press: 

Champaign, Cambridge, New York, and Melbourne, 1999.
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