Proceedings of DETC'06 2006 ASME Mechanisms and Robotics Conference Philadelphia, PA, September 10-13, 2006
DETC2006-99661 AN AUTOMATED DESIGN SYNTHESIS METHOD FOR COMPLIANT MECHANISMS WITH APPLICATION TO MORPHING WINGS Hongqing Vincent Wang a R&D Engineer
David W. Rosen b* Professor
a
b
IronCAD Inc., 700 Galleria Parkway, Suite 300, Atlanta, GA 30339 The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology Atlanta, GA 30332-0405
* Corresponding Author: 404-894-9668,
[email protected]. attack angle, chord length, camber height, and so on [1]. The relation between the airfoil geometry and its performance is shown as Figure 1. Most airplane wings are sufficiently rigid without significant movement or twist during flight. For example, large aircraft wings designed for efficient high-speed flight incorporate some form of rigid trailing edge flap and perhaps a rigid leading edge device such as a slat to achieve high aerodynamics performance [1]. However, future airplanes may fly like birds with flexible wings, so-called, morphing wings. Morphing wings may consist of a single element with sophisticated structures that can reconfigure their shape and adapt to changing flying conditions. These changes can affect the aerodynamics of the wing. A change in the geometry of the wing might be used to control flight, suppress flutter, reduce buffeting effects, and maximize fuel economy. Morphing wings might enable the design of multifunctional aircraft.
ABSTRACT An automated design synthesis method is developed to design an airfoil with a reconfigurable shape, which can change from one type of geometry to another. A design synthesis method using unit truss approach and particle swarm optimization is presented. In the unit truss approach, unit truss is used as a new unit cell for mechanics analysis of cellular structures, including lightweight structures and compliant mechanisms. Using unit truss approach, axial forces, bending, torsion, nonlinearity, and buckling in structures can be considered. It provides good analysis accuracy and computational efficiency. A synthesis method using unit truss approach integrated with particle swarm optimization is developed to systematically design adaptive cellular structures, in particular, compliant mechanisms discussed in this paper. As an example study, the authors realize the design synthesis of a compliant mechanism that enables an entire closed-loop airfoil profile to change shape from NACA 23015 to FX60-126 for the desired morphing wing. The nonlinear behavior of compliant mechanisms under large deformation is considered. The resulting design is validated by testing its robustness and considering nonlinearity. KEY WORDS Compliant Mechanism, Cellular Synthesis, Morphing Wing, Airfoil 1
Structure,
Geometry
Pressure
Performance
Figure 1 Relation between Airfoil Geometry and Performance An example problem, a morphing wing concept for AAI's Shadow [2], is proposed as an example study in this research. AAI's Shadow is a small Unmanned Aerial Vehicle (UAV) for information collection [3]. The flight range and endurance of UAVs are limited by their fuel storage capacity. It is greatly desired to increase the flight range and endurance without the addition of fuel. During a mission, as the fuel is burned, the total weight of the UAV decreases. Therefore, the wings’ working condition changes, and a different airfoil shape would better serve the aircraft. The airfoil geometry is desired to adapt to the changing working condition for improved airfoil performance. Wings with adaptive shapes can minimize drag and improve the fuel
Design
INTRODUCTION
The aerodynamic performance of an airfoil greatly depends on the airfoil geometry. The distribution of pressure over the airfoil is highly influenced by the airfoil geometry, including 1
Copyright © 2006 by ASME
truss approach is presented. In Sections 3~6, the authors discuss the design of a morphing wing for the AAI Shadow UAV through problem formulation, design synthesis, nonlinearity consideration, and validation steps. Finally, the authors present a conceptual design of a morphing wing for the AAI Shadow UAV.
efficiency. In the AAI’s Shadow example studied by Gano and Renaud, the wing cross-section morphs from the NACA 23015 airfoil to FX60-126 as shown in Figure 2. The NACA 23015 airfoil, represented by the outer profile in Fig. 2, is bulky and has more capacity to store fuel at the beginning of the mission. The FX60-126 airfoil, represented by the shaded region, is slender and represents the shape at the end of the mission. The profile coordinates of NACA 23015 and FX60-126 airfoil cross-sections were obtained from UIUC airfoil data site [4]. The coordinates can be scaled uniformly. The chord length of both airfoils is 300 mm.
NACA 23015
2
OVERVIEW OF DESIGN SYNTHESIS WITH UNIT TRUSS APPROACH
In this section, the authors introduce a new design synthesis method with the unit truss approach. Our unit truss approach is a new approach for the mechanics analysis of cellular structures, including both lightweight structures and compliant mechanisms. In this paper, it is integrated with particle swarm optimization to realize the design synthesis of a compliant mechanism for a morphing wing.
FX60-126
2.1
Unit Truss Approach for Mechanics Analysis A few structural analysis approaches have been developed to analyze compliant mechanisms. Typical methods include the ground truss (discrete) approach and the homogenization (continuum) method. The ground truss approach can only provide a rough estimate for the geometry of designed structures [11-13]. The homogenization method using artificial unit cells provides better results, but it may result in nonrealizable elements and can be computationally expensive [14, 15]. The unit truss, a new unit cell, is proposed to analyze cellular structures. A unit truss consists of the central node and a set of the half-struts that are connected to that node. Every two neighboring unit trusses share a common strut. An example of a unit truss is shown in Figure 3. This new unit cell approach was developed to accurately and efficiently analyze lightweight structures and compliant mechanisms, and to support their systematic design. As shown in the stress plot of a sample unit truss in Figure 4, the strain and stress around the nodes are usually complicated due to considerable inter-strut interactions and large bending moments [16]. The unit truss is leveraged from the ground truss approach and the homogenization method. A microstructure cell is used to represent the material distribution in the homogenization method. An advantage of the unit truss is that it can be used as both the cell primitive for analysis as well as synthesis and is manufacturable, whereas the microstructures for homogenization are artificial and not manufacturable.
Figure 2 Airfoil Morphing from NACA 23015 to FX60-126 With the assumption of linear fuel consumption over time and constant propeller efficiency, Gano and Renaud computed the range and endurance for both variform and NACA 23015 airfoils [5]. The variform airfoil linearly morphs its shape from NACA 23015 to FX60-126, while the NACA 23015 airfoil stays static without shape change. Gano and Renaud concluded that the range of the UAV with the variform wing was 22.3% farther and the endurance was 22.0% longer than the initial static NACA 23015 airfoil [5]. Thus, the morphing wing airfoil has better performance than the static airfoil since it provides better fuel efficiency to fly farther. Unfortunately, the feasible structure models and appropriate materials to transform the shape change of fuel bladders into the shape change of the airfoils have not been sufficiently studied [5]. Many different mechanisms could potentially generate such a shape change. Quite a few morphing wing designs utilize smart actuators and materials, such as lightweight piezocomposite and shape memory alloys [6, 7]. However, most of those mechanisms are neither able to cause large scale effects, nor cost efficient [8]. Some researchers introduced the use of compliant mechanisms to realize shape morphing [9, 10]. A compliant mechanism changes shape through structural deformation, which is independent of the problem scale [8]. However, to the authors’ knowledge, no compliant mechanism has been systematically designed for the entire closed-loop morphing airfoil, as shown in Figure 2 (although they may be capable of doing so). Existing design methods have been demonstrated only on the upper or lower section of a morphing airfoil [8, 9]. More importantly, the nonlinearity of compliant mechanism deformation has not been sufficiently studied. This paper presents a design synthesis method that utilizes a new unit truss approach to develop a mechanics model for compliant mechanism analysis and design. We capture the nonlinear deformation behavior in our model that is inherent in the large shape changes necessary for morphing wings. In Section 2, an overview of design synthesis method with unit
Unit Truss 1
Unit Truss 2
Unit Truss 3
Figure 3 Definition of Unit Truss
2
behavior of compliant mechanisms. Linear elasticity theory is used to solve nonlinear problems. The behavior of an elastic unit truss can be traced back incrementally using Equation 4, which is in a linear form [17]. K t is the tangent stiffness
u1(2) u1(1) u
u2(2) u3(1)
(1) 2
u1(5)
u
u1(0) u3(5)
u3(2)
(3) 1
u
u3(0)
u2(5)
matrix, dU are the incremental nodal displacements, and dF are the incremental nodal forces. Both geometric and material elastic nonlinearities are considered using Equation 5. The linear elastic stiffness is designated as K e , while K g and K m
u2(3)
(0) 2
u3(3)
u3(4)
represent geometric nonlinear stiffness and material nonlinear stiffness, respectively [17].
u1(4)
u2(4)
Figure 4 Using Unit Truss
unit truss, while U and F represents the nodal displacements and forces. Unit trusses can have any number of incident struts. They can be thought of as special elements with which to analyze large cellular structures using methods similar to conventional the finite element method. The geometric interactions between struts at nodes are adjusted by correcting the diagonal component in the stiffness matrix K e .
[U ] = u~ (0)
u (1) ~
(N ) Φ12 0 0 (N ) Φ 22 3( N +1)×3( N +1)
0 Φ (2) 22 0 u ( N ) ~
Nodal forces:
[ F ] = f (0) ~
f (1) ~
f (N ) ~
5
A Design Synthesis Method Using Particle Swarm Optimization as Search Algorithm A design synthesis method using particle swarm optimization was developed to systematically design adaptive cellular structures. The design synthesis of adaptive cellular structures is a large-scale nonlinear problem with multiple objectives and a mixed-discrete design space. The design problem formulation for compliant mechanisms is shown in Figure 5. Strut diameters are the design variables represented by xi (i = 1, 2,…, n), where n is the number of struts in the starting structure topology. The constraints include the bounds on strut diameters, static equilibrium, and stress. The objectives are to minimize the normalized Mean Squared Deviation (mean( SDk )) norm between the desired shape and the actual
1
(2) Φ12
Kt = Ke + K g + K m
2.2
Stiffness:
N (i ) (1) ∑ Φ11 Φ12 i =1 (1) Φ (1) Φ 21 22 [ K e ] = Φ (2) 0 21 (N ) 0 Φ 21 Nodal displacements:
4
Compared to the ground truss approach, our unit truss approach provides better accuracy to analyze compliant mechanisms by simultaneously analyzing multiple-degree-offreedom deformation and considering nonlinearity. Compared to the homogenization method, the unit truss approach is more efficient due to fewer microstructures used for analysis. As a result of this research, the analysis of lightweight structures and compliant mechanisms can utilize the same mechanics model (unit truss). This enables us to design structures for both stiffness and compliance simultaneously.
The mechanics model of unit trusses was successfully developed to analyze conformal cellular structures. The model includes considerations of axial forces, bending, torsion, nonlinearity, and buckling [16]. The constitutive equations of 2-D and 3-D unit trusses are derived using beam theory as shown in Equations 1~3. K e denotes the linear elasticity of a
Static equilibrium: K e ⋅ U = F
K t idU = dF
2
shape under deformation, and to minimize the normalized total material volume Vnorm. Detailed explanation is given in the next section. Particle Swarm Optimization (PSO) and Genetic Algorithms (GA) were selected from available optimization algorithms to systematically search for design solutions. PSO simulates the movement of birds in a flock, where individuals adjust their flying according to their experience and other individuals’ experiences during searches for food [18]. It combines local search with global search. PSO shares similarities with GA. However, PSO enables cooperative behavior among individuals (“birds”), as well as the competition modeled using GA. Hence, PSO often converges more quickly than GA and was selected for the design synthesis of cellular structures [19].
T
3
T
Geometric and material nonlinearities are considered with the tangent stiffness method. Geometric nonlinearity occurs in structures undergoing large displacements or rotations, large strains, or a combination of these. Material nonlinearity occurs due to nonlinear stress-strain behavior. The tangent stiffness method is a linearization approach to analyze the nonlinear
3
Find: Strut widths - x = {x1 , x2 ,
actual deformed profile shape is the shape of the source profile (NACA 23015) under deformation. The design objective is to approximate the actual profile shape to the target profile shape as close as possible. The desired deflections are measured from the corresponding points in NACA 23015 and FX60-126 airfoil profiles, and given as Equation 8.
, xn }
Satisfy: Bounds: xi ∈ 0, [ xmin , xmax ]
{
}
Constraints: static equilibrium - h1 : Kt idU = dF ,U |xi =0 = 0, F |xi =0 = 0
N33
N69
stress - g1 : σ (x) ≤ σ max
y
N8
f ( x, U ) = w1 × (mean( SDk )) norm + w2 × Vnorm x
wk × vi velocity inertia
social behavior
δ8
target
δ 28
NACA 23015 Source profile
N16
N 63
FX60-126 Target profile
target
= 6.58, δ 12target = 11.4, δ16target = 13.7, δ 24target = 1.56
8
= 2.14, δ 33target = 0.0, δ 63target = 9.43, δ 69target = −3.65
Even though the airfoil’s angle of attach changes during a mission, the change is not a shape change, but a rigid body rotation. The change of the attack angle can be realized by rotating the entire airfoil around the profile normal. Therefore, the relative positions of some specific points on the profile are assumed to be fixed in certain directions. For simplicity, the left-most points of the profiles of both NACA 23015 and FX60-126 airfoils are coincident. The left-most point P0 is fixed in both the x and y directions. The most right point P1 is fixed in the y direction, and able to move along the x direction. Therefore, there are totally 3 constraints applied on the airfoil profile, which is fully constrained in the profile plane when considered as a rigid body. The upper and lower sections of the profile are pin-jointed with each other at points P0 and P1 during deformation. The design synthesis objective consists of two parts. The first part is the normalized mean squared deviation (mean( SDk )) norm among all sampled nodes Nk (k = 8, 12, 16,
6
xi = xi + vi 7 where, wk = velocity inertia weight, ϕ1 is the cognition learning factor, ϕ2 is the social learning factor, rand() generates a random value in the range [0, 1], pi is the best position of particle i, and pg is the best position of any particle. The authors developed the design synthesis method by integrating PSO with the unit truss approach to systematically design compliant mechanisms with multiple inputs/outputs. This method is capable to support the design of compliant mechanisms for morphing wings. 3
N12
Desired Deflections:
+ ϕ1 × rand () × ( pi − xi ) + ϕ 2 × rand () × ( pg − xi ) cognition behavior
P1
Figure 6 Known Boundary Condition and Sampling Points for Design Synthesis
The PSO algorithm will be presented conceptually. Each of the p particles (birds), which represent design instances, is given an initial position and velocity. In our case, an initial position corresponds to a set of strut diameter values for the entire mechanism. The velocity is the rate of change of position (or strut diameters). The key step in the PSO algorithm is how to update the velocity from one iteration to the next. The new velocity is given as a combination of the current velocity, a velocity change based on the particle’s learning, and a velocity change based on the flock’s behavior (social learning) as given in Eqn. 6. Positions are computed simply using Eqn. 7. During each iteration, all particles’ velocities and positions are updated, then the objective function is computed. Convergence is checked by considering changes in objective function values, as well as design variable changes. vi =
N 24
P0
Minimize: mean squared deviation and material volume
Figure 5 Problem Formulation of Compliant Mechanism Design
N 29
24, 29, 33, 63, 69). mean( SDk ) is a statistical indicator to measure the closeness between the actual deformed profile and the target profile. As shown in Equation 9, the squared deviations are defined as the squared values of the differences between the actual deflections δ kactual and the target deflections
δk
PROBLEM FORMULATION FOR MORPHING WING DESIGN
target
on the measured nodes. (mean( SDk )) norm
is the
normalized value, which is the ratio between mean( SDk ) and the mean squared deviation of an initial guess before synthesis, mean( SDk ) NoSyn , as shown in Equation 10. Certainly,
The goal of the design synthesis is to design a compliant mechanism that can drive the airfoil shape to morph from the source profile (NACA 23015) into the target profile (FX60126) under a specific loading condition as shown in Figure 6. This compliant mechanism must fit in both profile shapes. The
mean( SDk ) NoSyn is not necessarily a precise value, but an
4
estimated value. For this particular problem, mean( SDk ) NoSyn is set as 60.0. The other part in the design objective represents the goal to minimize the total volume of the structure. This goal does not contribute to minimize the deviations, but it can help remove unneeded material from the resulting structure. A normalized volume Vnorm, the ratio between the volume of the current structure and that of the first run in the search process as shown in Equation 11, is used as the second part in the design objective. The contribution of the second part is much smaller than the mean squared deviation. It noticeably contributes to the design objective at a later stage of the search process. These two parts are compromised in the design objective using weights that model their relative importances. Their weights were selected as wd = 60.0 , wv = 5.0 . Their
vertical forces at nodes, N 36 , N 44 and N 55 . However, the objective function values resulting from these first two trials (28.1 and 19, respectively) are significantly larger than the third trial discussed as following. To better propagate the deformation within the structure, five pairs of equal and opposite forces are applied at the following nodes: N 4 and
N 69 , N8 and N 40 , N12 and N 30 , N16 and N 26 , N19 and N 23 in the third trial. The forces on these five pairs of nodes are all set as Fpair = 21N in magnitude, but each pair of forces are in opposite directions shown by the green arrows in Figure 9. The resulting deformation before synthesis is unnoticeable.
ratio, wd / wv = 12.0, indicates that the mean squared deviation is much more influential than the volume on the design objective. The mean squared deviation is dominant at the early stage during the search process. Squared Deviation: SDk = (δ k − δ k ) actual target
2
Normalized volume: Vnorm =
interior struts
– xi ∈ x0+ , xmax
i = 2 ~ 59,88,89,91 ~ 94,99 ~ 104,106,113,114,116 j = 1,60~87,90,95,97,98,105,107~112,115,117 Minimize: f ( x) = wd × (mean( SDk )) norm + wv × Vnorm
10
mean squared deviation
volume reduction
where, wd = 60.0 , wv = 5.0 , k = 8,12,16, 24, 29,33, 63, 69
Vtotal VFirstRun
Figure 7 Problem Formulation of Compliant Mechanism for Variform Wing
11
The design synthesis process starts with an initial topology as shown in Figure 8. The initial topology has a total of 72 nodes and 117 struts. For the convenience for applying boundary constraints and loads, 2 separate nodes (N1 and N72) are created at point P0. These 2 nodes are fixed in the x and y directions. N1 is one node of strut 117 in the upper section, and N72 is one node of strut 1 in the lower section. Similarly, 2 nodes (N2 and N71) are created at point P1. N2 is for strut 95 in the lower section, and N71 is for strut 115 in the upper section. The problem formulation of the design synthesis for the variform wing is shown in Figure 7. The design variables are the diameters of the struts in the initial topology shown in Figure 8. The design objective is to minimize the weighted sum objective value of the mean squared deviation, and the normalized volume. 4
Satisfy: Bounds:
, xn } Widths of lattice struts
where, x0+ = 2.5e − 4, xb min = 5.0, xmax = 8.00
9
mean( SDk ) mean( SDk ) NoSyn
x = {x1 , x2 ,
boundary struts – x j ∈ [ xb min , xmax ]
Normalized Mean Squared Deviation:
(mean( SDk ))norm =
Find:
The synthesis result of the structure under five pairs of opposite forces is shown in Figure 10. The objective function value, f ( x ) = 4.367 , is significantly less than the above two trials. The important PSO parameters used are listed as: number of particles = 20, inertia weight ranged from 0.95 to 0.4, cognition learning factor = 2, social learning factor = 1.25, and maximum number of iterations = 150. Totally 10 runs were performed with 38300 evaluations of the objective function for 4.81 hours of CPU time. Each evaluation of the objective function takes about 0.612 seconds. The experiment computer has an Intel P4 2.4GHz CPU and 512MB RAM. The final objective function value of each PSO run is listed in Table 1. The average value is 12.18, which is significantly better than the other two trials under different load conditions. The best run (the 3rd run) has the objective function value, f ( x ) = 4.37 , and its deformed shape is shown in Figure 10.
DESIGN SYNTHESIS OF MORPHING WING
The authors have attempted three trials under different load conditions. The first trial is under a concentrated torsion at an interior node, N 45 . The second trial is under three concentrated
Table 1. Objective Functions of PSO Results Run NO.
1
2
3
4
5
6
7
8
9
10
Average
f(x)
10.71
13.25
4.37
11.00
21.34
10.75
10.41
9.96
9.65
20.39
12.18
5
(A) Nodes of starting topology
(B) Elements of starting topology
Figure 8 Initial Topology for Design Synthesis
f ( x) = 59.93 : mean( SDk ) = 58.85 , Vnorm = 0.2157 Figure 9 Deformed Shape with Five Pairs of Opposite Forces before Synthesis
f ( x) = 4.367 : mean( SDk ) = 2.854 , Vnorm = 0.3027 Figure 10 Deformed Shape with Five Pairs of Opposite Forces after Synthesis
(A) Before Deformation
f ( x) = 5.012 : mean( SDk ) = 3.690 , Vnorm = 0.2643 (B) After Deformation Figure 11 Cleaning Topology by Removing Zero-width Struts 6
13.000 11.000 9.000 7.000 5.000 Force/N
Force/N f(x)
13.0 9.100
14.0 7.347
15.0 6.126
16.0 5.548
17.0 5.779
18.0 7.089
19.0 10.041
20.0 16.059
Figure 12 Variations under Nonlinear Deformation
CONSIDERATION OF NONLINEARITY
In Figure 12, the authors show that the objective function changes when deformations include geometric nonlinearity considerations. The tangent stiffness method discussed in Section 2 is used for the nonlinear deformation analysis with 10 steps of linear analysis [20]. The nonlinear deformation analysis could be considered during design synthesis. However, the total computation time would be scaled up by the number of steps since only one step is used during linear deformation analysis. When the force is maintained as Fpair = 21N , the
7.000 6.000 5.000 4.000 18.0 19.0 20.0 21.0 22.0 23.0 24.0 Force/N
objective function is f ( x ) = 16.059 , which is much larger than that under linear deformation. The deformation shape shows that the structure is over-deformed. According to the chart shown in Figure 12, the objective function reaches the minimum f ( x) = 5.548 when Fpair = 16 N . The geometric
Force/N f(x)
18.0 6.116
19.0 5.498
20.0 5.129
21.0 5.012
22.0 5.144
23.0 5.528
24.0 6.162
Figure 13 Objective Function Changes against Load Condition Variations
Objective Function f(x)
nonlinearity makes the structure softer under large deformation. The resulting design is reliable against variations of the loads as well. Fpair = 16 N is proposed for the magnitude of the five pairs of nodal forces when considering geometric nonlinearity. With the consideration of nonlinearity, the designed mechanism can better approximate the desired shape due to the higher analysis accuracy. Nonlinearity analysis is recommended to refine the resulting design from design synthesis when only considering linear deformation. 6
15.000
13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0
Objective Function f(x)
5
Objective Function f(x)
A topology cleaning process is performed on the resulting synthesized topology by removing “zero-width” struts, of which the widths are close to 0.00025. During the cleaning process, the struts that have only one node connected to the structure are removed. The cleaned structure has 68 struts and 60 nodes as shown in Figure 11 (A) and (B). After cleaning, the objective function value f ( x ) = 5.012 is a little larger than that before cleaning. The change is caused by the removed struts that contribute to the structure’s stiffness even if their widths are relatively small. However, this does not significantly influence our design synthesis result. In Table 2, the authors show the deflection deviations of the sampled nodes between the actual profile and the target profile. The deflection deviations of the sampled nodes are significantly improved if compared to those before design synthesis.
20.000 15.000 10.000 5.000 0.000
VALIDATION
-0.20 -0.10 -0.05 0.00 0.05 0.10 0.20
This section discusses the robustness of the obtained result against the variations of the load condition and the strut diameters. Then, the difference of the structure’s performances with linear and nonlinear deformation analyses is presented. In Figure 13, the authors show that the objective function modestly changes against the variations of the load condition, which could be caused by the operating environment. When the force magnitude changes around the designated value, Fpair = 21N , by 4.76%, the objective function changes by
Thin Strut Width Width Deviation f(x)
-0.20 18.431
-0.10 17.259
-0.05 14.929
0.00 5.012
0.05 15.986
0.10 17.555
0.20 18.506
Figure 14 Objective Function Changes against Width Variations of Thin Struts The manufacturing process may induce variations of design variables. Variations of thin struts can incur large changes in the objective function as shown in Figure 14. A variation of thin struts diameter by 0.05mm can cause a 200% change in the objective function. As shown in Figure 15,
2.63%, which means a relatively small change.
7
from NACA 23015 to FX60-126. The nonlinear behavior of the deformed compliant mechanism is considered with the unit truss approach, since the deformations are large. Five pairs of concentrated forces acting in opposite directions are applied to the morphing wing. Their load magnitudes can be refined with a sub design synthesis to determine more suitable magnitudes. Furthermore, as part of future work, the load conditions could be considered as design variables during the design synthesis, instead of a follow-up refining process. The unit truss approach enables the analysis of 3-D structures and the presented design synthesis method has the potential to design 3-D compliant mechanisms. It could greatly advance the design synthesis method of 3-D compliant mechanisms. It could be applied to broader classes of problems, such as scaffold design in tissue engineering and novel material structure design.
Objective Function f(x)
variations of thick struts do not influence the objective function much. When the diameters of thick struts change by 0.1 mm, the objective function changes about 4.37%. Thus, variations of thick struts are not critical to the mechanism’s performance, while variations of thin struts are very influential. Improved methods for addressing this sensitivity remain an open issue for future research.
7.000 6.500 6.000 5.500 5.000 4.500 4.000 -0.20 -0.10 -0.05 0.00 0.05 0.10 0.20 Thick Strut Width
Width Deviation f(x)
-0.20 5.837
-0.10 5.231
-0.05 5.081
0.00 5.012
0.05 5.014
0.10 5.080
0.20 5.375
(A)
Figure 15 Objective Function Changes against Width Variations of Thick Struts 7
N 29
THE RESULTING CONCEPTUAL DESIGN
(B)
N12 NACA 23015 Source profile
x
N16
N 63
FX60-126 Target profile Fuel bladder working li Force pair
Figure 16 A Conceptual Design Using Multiple Fuel Bladders to Drive the Shape Change of Morphing Wing [5] ACKNOWLEDGEMENTS We gratefully acknowledge the U.S. National Science Foundation, through grants IIS-0120663 and DMI-0522382, and the support of the Georgia Tech Rapid Prototyping and Manufacturing Institute member companies for sponsoring this work.
In this paper, the authors presented a design synthesis method with the use of the unit truss approach and successfully utilized it to design a compliant mechanism for a morphing wing of AAI’s Shadow UAV. The resulting compliant mechanism enables the entire morphing airfoil to change shape
Table 2. Deviations between Actual and Target Deflections of Sampled Nodes 8 6.580
P1
(C)
CONCLUSIONS
Node ID Target
N 24
P0 N8
There could be various approaches to realize morphing wings. Gano and Renaud proposed to use fuel bladders, which interact with the structure of the wing [5]. The shape changes of the fuel bladders are used to drive the shape change of the wing airfoils. As fuel is consumed, the fuel bladders decrease in size. This size change is transferred to the wing to cause it to change shape, thus leading to a morphing wing capability. In this paper, we propose to use the same concept and offer the configuration shown in Figure 16 as one example. Multiple bladders of difference sizes and shapes pull the relevant nodes on the compliant mechanism to drive its shape change, as shown in Figure 16(A). Five of these fuel bladders are used as actuators, as indicated by the springs in Figure 16(B) and (C). 8
y
Fuel bladder working like springs
N 33
N 69
12 16 11.400 13.700
24 1.560
28 2.140
33 0.000
63 9.430
Actual -6.397 -11.360 -13.759 -1.686 -2.250 -0.020 -9.427 Before synthesized Deviation -12.977 -22.760 -27.459 -3.246 -4.390 -0.020 -18.857 Actual 5.226 11.515 13.532 1.250 3.125 3.031 6.848 After synthesized Deviation -1.354 0.115 -0.168 -0.311 0.985 3.031 -2.582
8
69 -3.650 3.645 7.295 0.225 3.875
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