Design of a Shape-Changing Rigid-

Body Parabolic Light Reflector

Honors Thesis

Mark M. Plecnik

Department: Mechanical Engineering

Advisors: Andrew P. Murray, PhD; David H. Myszka, PhD, PE

April 2010

Design of a Shape-Changing Rigid-

Body Parabolic Light Reflector

Honors Thesis

Mark M. Plecnik

Department: Mechanical Engineering

Advisor: Andrew P. Murray, PhD; David H. Myszka, PhD, PE

April 2010

Abstract The advantage of the ability to shape-change is that a system can dynamically manipulate its geometry in order to optimize for a trade-off situation over multiple scenarios. This is opposed to regular static optimization methods. In particular, shape-changing mechanisms composed completely of rigid links and classical mechanical joints can be designed through a recently developed kinematic synthesis procedure. This kinematic synthesis may provide solutions for a variety of applications including light reflection, solar concentration, ergonomic chair design, automobile aerodynamics, and acoustic horns amongst others. The focus of this thesis is the design process of a shape-changing parabolic light reflector from the inception of profile curves to the fabrication of a functional prototype.

Table of Contents

Abstract Title Page

1. Background 1

A. Description of Rigid and Compliant Materials 2

B. Advantages and Disadvantages of Rigid and Compliant Materials 4

2. Potential Applications 5

A. Solar Concentration 6

B. Ergonomic Seating 7

C. Acoustic Horn 9

D. Automobile Aerodynamics 9

3. Focused Application: Light Reflector 9

4. Design Process 12

A. Parabola Selection 12

B. Creating/Evaluating Mechanisms 16

C. Analyzing/Modifying Mechanisms 19

D. Designing a Prototype Mechanism 22

5. Resultant Prototype 28

6. Conclusion 29

Reference 30

Appendix A: Engineering Drawings 33

Appendix B: Purchased Parts 43

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1. Background

The concept of shape-change refers to the ability of a mechanical system to alter

geometries within a design in order to improve performance or add functionality. This

concept is also referred to as morphing or adaptive technology. The advantage of shape-

change is that a system no longer has to statically optimize a trade-off situation, but can

now dynamically manipulate that situation in order to achieve optimization for a

multitude of scenarios. These scenarios refer to a discrete or continuous set of inputs that

a design is most likely to encounter.

The typical example of a mechanical design that can benefit from shape-change is

the aircraft wing. Aircraft face a bevy of scenarios which dictate the necessity for shape-

change. For example, variable-sweep winged aircraft such as the Bell X-5 demonstrate

an attempt to improve aircraft performance over a range of velocities [1]. Velocity is

considered a continuous set of inputs as it can be broken down into an infinite amount of

divisions. The resulting shape-change would thus be infinitesimal in nature as well. On

the other hand, unmanned aerial vehicle (UAV) wing geometry can be dictated by a

discrete set of scenarios, such as reconnaissance or attack. Therefore, the resulting

geometry would consist of two target formations, one for each mode. Other potential

scenarios include hunt or rescue, and biological or nuclear weapon detection [2]. The

aircraft wing represents a fitting application for shape-change due to the presence of flow

fields. Flow field designs tend to be good applications as they are controlled by

geometries. Other geometry-centric areas of interest include reflection of

electromagnetic waves [3], reflection of acoustic waves, and ergonomics. Each of these

general applications can benefit from the dynamic optimization afforded by shape-

change.

This thesis will begin by explaining various methods of achieving shape-change

and provide a comparison of morphing via rigid or compliant materials. Chapter 2

presents several applications in which rigid-body shape-change may provide a viable

solution. Chapter 3 provides a description of the application and Chapter 4 describes the

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design of a prototype for the application from inception of the profile curves to the design

of a functional prototype. Chapters 5 and 6 present the results and conclusions.

A. Description of Rigid and Compliant Materials

The means to achieve shape-change can be broken down into two main research

efforts: (1) morphing through rigid bodies and (2) morphing through compliant materials.

However, there is a certain amount of gray area between these two sectors of study. For

example, a lumped compliance mechanism will employ links made of a rigid material but

also utilize thinned-out compliant sections to form flexural joints [4]. Although this

thesis discusses the advantages of each technology separately, it is important to keep in

mind that the best shape-change solutions may come from a combination of rigid and

compliant materials.

However, the focus of this discussion is the development of rigid-body

mechanisms. These mechanisms are defined by kinematic linkage systems consisting of

perfectly inflexible members connected by the classical joints of mechanical design i.e.

revolute joints, prismatic joints, etc. As well, these linkage systems can be categorized

by the number of degrees of freedom that they possess. In specific, this thesis focuses on

single degree of freedom mechanisms. The construction of rigid-body mechanisms

consists of two sets of links: (1) profile links and (2) dyads [5]. Profile links are shown in

Figure 1 as either blue or green. Dyads are shown in red. The profile links are defined as

the portion of the linkage used to approximate a set of target profiles for a given shape-

change application. These curves are properly oriented and joined at their endpoints by

revolute joints. A set of dyads connect the profile links to ground in order to reduce the

degrees of freedom. These mechanisms are typically actuated through the rotation of a

single dyad. An example of a rigid-body mechanism is shown in Figure 2. A shape-

changing cam was designed to transition between three generic cam profiles: a step cam,

an egg-shaped cam, and an off-axis circle cam. The example serves to illustrate the

capabilities of rigid-body mechanisms to create drastic shape-change.

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Figure 1. Kinematic layout of a closed chain shape-changing mechanism used to create three cam profiles: step, egg, and off-center circle.

Figure 2. A shape-changing cam derived from the kinematic layout of Figure 1.

The other means to shape-change are compliant materials. Compliant materials

consist of a wide array of technologies from simple elastic materials to intricate smart

materials. Smart materials are an advanced technology characterized by their ability to

transition between rigid and flexed states through various means of distributed actuation.

Distributed actuation refers to the application of heat [6], a magnetic field [7], or an

electric current [9] across the area of a material in order to actuate shape-change.

Furthermore, there are also less common means of distributed actuation such as light

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exposure [9]. One of several examples of smart materials are shape memory polymers

(SMPs). SMPs are a thermally actuated morphing technology defined by their glass

transition temperature. They are rigid below their glass transition temperature and

become pliable above. SMPs have the ability to become heated, shaped, then cooled in

order to achieve a new rigid shape. Upon the re-application of heat, an SMP will then

transition back to its original shape [6].

B. Advantages and Disadvantages of Rigid and Compliant Materials

There are several key differences between rigid and compliant mechanisms that

illustrate the strengths and weaknesses of each technology. These differences include the

complexity of development, surface accuracy, seamlessness, and extent of shape-change.

As both technologies are still in their research stage, the effort required to create

an initial design for both a rigid-body mechanism and a compliant mechanism is

substantial. However, once a design is advanced into development stages, a disparity

emerges between the complexities of these technologies. After the initial kinematic

synthesis of a rigid-body design, a practical mechanical design can be developed. As

stated earlier, these designs are made up of classical mechanical components which tend

to yield a basic and well understood design problem. Furthermore, simple rotary or linear

actuators are all that is required to produce the shape-change motion. On the other hand,

designs involving the distributed actuation of a smart material will often require complex

thermal [6] or electromagnetic [7] means to do so. Along these lines, there are several

aspects of smart material morphing technology that are currently very developmental in

their nature, most notably, the materials themselves and the relationship between the

materials and means of actuation [10].

Surface accuracy refers to the exactness in which a morphing technology can

produce the desired surfaces associated with a given set of target profiles. Due to their

nature, rigid-body mechanisms are inherently limited in this respect, as rooted by the fact

that multiple target profiles are approximated by a single finite set of curves which

determine the profile links. This approximation will always involve a certain amount of

error except in a few rare cases. Therefore, rigid-body mechanisms alone may not be the

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optimal solution for precision applications. On the other hand, a smart material utilizing

a versatile form of distributed actuation has a greater potential for high surface accuracy

and fine adjustment, for example, in the shape control of a space satellite’s reflective

surfaces [10]. Moreover, as joints are excluded, these technologies can produce smoother

contours. Conversely, although rigid-body mechanisms may be less apt for precision

applications, they are better equipped to create more sweeping shape-change.

It is also important to note that mechanisms combining rigid and compliant

materials could potentially take advantage of the strengths of both. A design might

utilize the drastic shape-change and load-bearing capabilities of rigid-body mechanisms

combined with the precision capabilities of smart materials. Specifically, this approach

could prove useful for a morphing airfoil. A major design challenge with this approach is

compensating for the mechanical impedance mismatch between rigid and compliant

materials at their interface locations [6].

2. Potential Applications

Applications that are suitable for single degree of freedom rigid-body shape-

change technology, hence forth referred to as shape-change technology, tend to be

characterized by a specific set of needs. These needs include a large degree of profile

change, a lenient precision requirement, and a sufficient benefit for optimizing for a

variable input or generating a variable output. The applications researched in this paper

illustrate how the presence or absence of these characteristics can either encourage or

discourage the implementation of shape-change technology. These applications include

solar concentration, chair ergonomics, acoustic customization, and automobile

aerodynamics.

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A. Solar Concentration

Solar concentrators make use of some basic geometric definitions in order to

direct rays into a focus. Most notably, the parabola is utilized due to its ability to reflect

all rays parallel to its axis into a central focus. Elongated parabolic troughs and

paraboloid dishes are two technologies that utilize this geometry. Parabolic solar

concentration systems must compensate for the changing angle of incidence of the sun’s

rays throughout the day in order to keep its axis parallel. These rays represent a variable

input with the ultimate goal to maximize the output at the focus. However, shape-change

technology would be hard pressed to provide a solution better than simply rotating a

trough or dish about one or two axes as is currently done. Shape-change is not required

in order to follow the sun. Furthermore, the current reflectors deal with a level of

precision greater than one that could be provided by shape-change technology at this

point.

However, there are solar concentration design problems with a less strict precision

requirement that may be more geared toward shape-change. Wolfgang Scheffler has

identified a need in third world countries for cooking through solar concentration. The

basis of his idea is to move the focus to a fixed location out and away from the parabolic

reflector. This would allow a reflector outside a building to direct energy to a focus

inside the building where it could be used to cook. As the focus is fixed, the reflector

must move in order to compensate for the sun’s changing angle of incidence. Scheffler

accomplishes this for both daily and seasonal change through polar rotation and flexing

the parabolic dish, respectively. Additionally, he is able to create a sufficient amount of

precision for his application through basic manufacturing methods [11]. The opportunity

for shape-change comes in the form of replacing the flexible reflector with a rigid-body

mechanism. Furthermore, this concept does not need to be restricted to solar cookers.

Rigid-body Scheffler-like reflectors could be incorporated into a heat engine based

generating system such as those associated with Stirling engines. Current Stirling engine

solar power systems do not incorporate a fixed focus and therefore require a large boom

inside the reflector to hold the Stirling engine unit at the paraboloid’s focus [12].

However, the institution of Scheffler-like reflectors could remove the boom and place the

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Stirling engine on the ground where it would no longer be constrained by shape or size.

An example of potential Scheffler curves is shown in Figure 3. The major concerns with

incorporating shape-changing technology into this application are that a lack of drastic

shape-change may prove another technology to be more suitable and that surface

accuracy issues may arise.

Figure 3. Potential Scheffler parabolic curves. The open circles represent the location of

the vertex on each curve.

B. Ergonomic Seating

Within the field of ergonomics, the idea of a lumbar supporting shape-changing

seat provides a plausible concept. Sitting for long periods of time can often cause back

pain. This is a concern for people who spend a lot of their time in a driver’s seat or an

office chair. Specifically, a lack of proper support for the lumbar region of the spine can

produce high intradiscal pressure which could lead to a herniated intervertebral disc. The

objective is to take the spine out of a flexion position and move it into its natural lordotic

curvature. A study from the British Osteopathic Journal which measured the seated and

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standing spinal profiles of 20 individuals revealed all subjects to move into a flexion

position when seated in a typical car seat [13]. However, the amount of lordotic

curvature needed from person to person varies. Past studies have indicated a spatial

adjustment range of a lumbar support on the order of 60 mm to be necessary for the

proper accommodation of a sample of male and female subjects [14].

The advantage of a shape-changing seat would be its ability to accommodate a

large range of spines. Figure 4 and Figure 5 demonstrate two designs for a car seat and

office chair application, respectively. The profiles were generated from the

measurements published in the British Osteopathic Journal. The design process used to

create each mechanism was similar to the one described in Chapter 4.

Figure 4. Shape-changing car seat.

Figure 5. Shape-changing office chair.

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C. Acoustic Horn

An application in the field of acoustics may be a shape-changing horn. Horns

serve many functions from their use in musical instruments to material handling

processes. They are also prevalent in the design of loudspeakers as a means to efficiently

transfer sound from a driver to the open air. The design of a loudspeaker horn is very

much centered on its shape. Horns are usually manufactured to form an exponential

geometry for high frequency drivers but can also be found in other shapes i.e. elliptical,

hyperbolic cosine. The geometry of a given horn is dictated by the driver that it is

enhancing, listener preferences, and the surrounding acoustic environment. Thus,

altering these variables would necessitate an altered geometry. Bangtsson et al. discuss a

method for optimizing horn shape for single or multiple frequencies [15]. Shape-change

could potentially be applied to these results.

D. Automobile Aerodynamics

The automobile presents several opportunities for shape-change, most notably,

aerodynamic optimization. For example, the shape of the rear end of a car is often the

result of a compromise between drag and rear lift [16]. However, shape morphing can

allow the rear end to optimize for multiple scenarios through the manipulation of the

drag/lift trade-off. Bringing this concept to a more radical level, a car could potentially

also change body types from an aerodynamic coupe to a cargo-friendly hatchback. This

application would likely include a combination of rigid and compliant materials.

3. Focused Application: Light Reflector

The application of interest in this thesis is the manipulation of a point light source

through a parabolic reflector. The idea is that if the reflective panels used to direct light

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in many fixtures could change their shape, they could then optimize the distribution of

light for various scenarios. Specifically, a shape-changing light reflector would have the

ability to optimize the trade-off between intensity and spread of the light, while holding

the energy consumed by the light bulb constant. A parabolic light reflector essentially

utilizes the reverse process used by a parabolic solar concentrator. The objective of a

solar concentrator is to receive parallel rays and focus them to a point, however, a light

reflector receives a point source and collimates the light rays. As the light reflector is the

inverse of the solar concentrator, it turns out that this characteristic makes it a more

suitable application for shape-change. Whereas the objective of solar concentrators is to

maximize the output, the objective of light reflection is to create a variable output. The

fact that the output for light reflection requires multiple levels serves to better justify the

use of shape-change mechanisms for this application as more drastic shape-change is

required to produce these levels.

The basic layout of a shape-changing parabolic light reflecting system would very

much resemble that of a trough concentrator. A schematic is shown in Figure 6. A

reflective parabolic surface is extended lengthwise, creating a focal axis above itself.

Along this focal axis is placed a tubular light source. Behind the reflector is all of the

mechanism necessary to complete the desired shape-change. The desired shape-change

would consist of the reflective panels forming an array of different parabolas with respect

to the fixed focal line. The light bulb and mechanism would be held in place by

mounting plates.

Figure 6. Schematic of shape-changing light reflector.

Light Source

Reflective Surface

Mechanism Mounting Plates

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Adaptive lighting has the potential to be useful in a variety of functions. One

such use may be task-ambient lighting. Task-ambient lighting refers to an energy

efficient method of lighting large areas, such as office spaces, in which ambient

illuminance is reduced while task illuminance maintained. Current methods are based on

the removal of luminaries which had the primary purpose of general area lighting such as

large overhead fluorescent tubes commonly found in most offices [17]. However, the

implementation of lighting fixtures capable of transitioning between large-area-spread-

illuminance and high-intensity-task-illuminance, all while holding power consumption

constant, would provide more flexibility to this technique. The advantage for an office

manager or general user would be the ability to control the actual light paths traveled as

opposed to only controlling intensity through a dimmer configuration or removing the

luminaries all together.

This same concept can also be brought to a micro level in which an individual

user could control the spread and intensity of his personal task lamp for a specific task.

For example, if the user is working on his computer he may only want a low intensity

spread around him. However, if he is reading a book, he may want a higher intensity

beam concentrated on where he is looking.

Another potential function for the light reflector concept is the integration into

automobile headlamps. Adaptive automobile lighting is a technology which has gained

popularity in recent years. The main idea is to improve visibility during nighttime

driving, when the rate of accidents and fatalities is greatest. This is accomplished by

dynamically adjusting the distribution of light based on various traffic and environmental

conditions. In fact, a European Economic Commission has developed patterns for low-

beam adaptive lighting according to town, country, motorway, and adverse weather

scenarios. Furthermore, the German company Hella KGaA Hueck has developed a

Vario-Xenon headlight capable of creating six different light distributions [18]. These

distributions differ based on the direction, spread, and intensity of light. Therefore, the

implementation of a shape-changing reflector can potentially be advantageous to the

industry as well. However, the specific mechanism presented below would most likely

be insufficient. A shape-changing automobile headlamp may need to contain asymmetric

and elliptical configurations.

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4. Design Process

The design process used in the creation of the shape-changing parabolic reflector

required several steps and a variety of software tools. These steps are displayed in Figure

7. First, the process consisted of choosing the proper shape-change to complete. Design

profiles were generated through the creation of an Excel worksheet to quickly graph a set

of parabolas. The next step was to create the mechanism necessary to perform the

desired shape-change. A Matlab program was used to produce a general layout of this

system. The design then needed to go through the iterative process of simulating,

analyzing, and redesigning. Most of this work was accomplished using Working Model,

a 2D motion simulation program. From there, a full prototype was designed using

AutoDesk Inventor and the prototype was fabricated at a local machine shop.

Figure 7. Flowchart of design process used in this thesis.

Creating/ Evaluating

Mechanisms

Analyzing/ Modifying

Mechanisms

Designing a Prototype

Mechanism

Fabricating the

Prototype

Choosing the Proper Parabolas

A. Parabola Selection

The selection of the proper parabolic curves to include in the shape-change

involved choosing values for a set of input parameters in order to generate output

parameters in accordance with an effective shape-change. The input parameters included

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h, k, and p as defined in the generic equation for a parabola where (h, k) are the

coordinates of the vertex and p is th vertex to the focus. e distance from the

4

A second set of input parameters included the range in order to define which section of a

parabola would be used.

The output parameters of the set of parabolas chosen for shape-change include

overall length, aperture length, and the ratio between these two numbers. The overall

length parameter was kept constant for each parabolic section chosen since rigid-body

shape-change assumes no deformation of the members involved. The length, s, of a

given parabola can be calculated using the generic equation for arc length based on a

range specified by x1 and x2.

1

After plugging in the parabola equation and performing the necessary integration steps,

the expression simplifies to the following formula.

12 1

14

12 1

14

Another important parameter of the chosen parabolas is the aperture length. The

aperture length is a proportional measurement of the resulting aperture area that a given

parabolic section would yield. Aperture length is defined by the range of a parabolic

section.

| |

Since aperture length, la, is used to eventually calculate the aperture area, Aa, of a

parabolic section extended by width, wa, and since aperture area is inversely proportional

to the resulting intensity of the light, I, therefore aperture length is also inversely

proportional to intensity through the following relationship.

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Therefore, the aperture length could be used to indicate the percent change of intensity

from one parabolic section to the next.

A final output parameter is the ratio between overall length to aperture length, R.

This dimensionless parameter is an indicator of how efficiently a parabolic section

consumes space and material. This parameter is highly dependent upon the curvature of

the parabola as dictated by the parameter p, as well as the range of the parabola. A lower

p value (narrower aperture) will produce a greater ratio. This ratio will always be greater

than 1. A value close to 1 indicates an efficient design. Greater values indicate that a

design is likely to have issues with surface accuracy. Figure 8 shows two asymmetrical

parabola sections with incident light ray paths. These sections have equal apertures

which means they are collimating the same amount of light energy. The section on the

right is not ideal.

Figure 8. The left parabola has a s/la ratio of 1.11. The right parabola has a s/la ratio of 2.17.

Aperture Length

Overall Length

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The parabolas chosen for shape-change are graphed in Figure 9. Table 1 shows

their input and output parameters. The data below indicates that there will be a decrease

in aperture length of 43% from Parabola 5 to Parabola 1. This means an increase in

intensity of that factor is possible, granted a mechanism could achieve perfect surface

accuracy. The s/la ratio was kept below 2.00 for all curves.

Figure 9. The parabolic design profiles chosen for shape-change.

Table 1. Parameters of the parabolas from Figure 9.

Color h k p s la  s/laParabola 1 Red 0 -0.5 0.5 12.0 6.22 1.93

Parabola 2 Yellow 0 -1.0 1.0 12.0 8.08 1.49

Parabola 3 Green 0 -1.5 1.5 12.0 9.18 1.31

Parabola 4 Blue 0 -2.5 2.5 12.0 10.38 1.16

Parabola 5 Purple 0 -3.5 3.5 12.0 10.98 1.09

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B. Creating/Evaluating Mechanisms

The next step of the process is to create and evaluate mechanisms in order to

complete the desired shape-change. This is facilitated through a Matlab based program

which automates much of the kinematic synthesis. The ShapeChanger program was

originally developed by Murray et al [5]. It completes two main critical tasks of the

design process of these mechanisms: (1) segmentation and (2) mechanization.

Segmentation refers to the approximation of a set of target profiles by a set of

rigid links. The first step is to transform the previously defined design profiles into points

defined by a set of piecewise linear curves known as target profiles. Mean profiles are

then generated across the target profiles through an error-based method to form the

segments that will become the rigid links. Once an acceptable error is specified, the

process is carried out through growing the mean profile segments one point at a time until

the acceptable error is surpassed, at which point a new mean profile segment begins and

the process continues until enough segments are generated to approximate the entirety of

the target profile. At this point, the error is then further reduced through moving

segmentation points in an iterative process until error ceases to decrease. This helps to

shorten the length of segments where shape-change is most dramatic and distribute error

more evenly [5].

The mechanization phase begins by joining each of these newly formed segments

at their endpoints with revolute joints to form a chain. This process tends to increase the

error as segments must be repositioned in order to do so. Next, dyad links are added

from the chain to ground until the degrees of freedom of the mechanism have been

reduced to 1. For an open chain, this involves adding a single dyad to each segment

except one, which receives two dyads. The set of one profile link, two dyad links, and

ground is known as the four-bar sub-linkage. It is a sub-mechanism that is determinate

by itself and allows the rest of the mechanism to be determinate as well. Next, the

location of the circle and center points for each dyad is constrained to particular areas as

specified. The center points refer to the locations of each dyad’s fixed revolute joint.

The circle points refer to each dyad’s moving revolute joint. Within the constrained

areas, circle and center points are selected at random for each dyad until a mechanism is

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arrived upon which produces the desired motion and is devoid of singularities throughout

the significant portion of its stroke. These mechanisms are considered successful designs

[5].

The Matlab-based process described above will generate zero successful designs

if a given shape-change is not solvable. On the reverse, if a given shape-change has a

multitude of solutions, then it can generate thousands of designs. In the case of the

parabolic light reflector, this number is closer to the latter. However, there are a

multitude of design criteria that are used to eliminate the extraneous solutions as listed

below.

1. Mechanism accurately reproduces design profiles.

2. Mechanism does not interfere with path of light.

3. Dyads are of reasonable length.

4. Mechanism involves a reasonable number of links.

5. Links do not excessively overlap.

6. Mechanism exudes a degree of symmetry.

Of the greatest importance is that a design is able to accurately reproduce the

desired design profiles. Figure 10 provides a sample of a mechanism that fails to meet

this as well as other requirements. The next most important criterion is that the generated

mechanism does not interfere with the light ray paths between the source and reflectors.

Therefore, it is best if all the mechanism is contained behind the profile links. Next, the

length of the dyads added during the mechanization phase must be within a certain range.

If these dyads are too short, they are subject to bearing great forces in near singularity

situations. If these dyads are too long, they become unwieldy and subject to bending

moments. The number of links and the degree to which they overlap throughout their

motion determine the practicality of a mechanism. The issue of overlap can usually be

dealt with by redirecting the path of dyad links so that they avoid interference with other

links. However, this fix becomes cumbersome when used too frequently in a given

design. Finally, from an intuitive standpoint, the appropriate solution for this specific

shape-change application should possess a degree of symmetry as this is a characteristic

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of the design profiles. The preliminary selected design as generated by the ShapeChange

software is shown in Figure 11.

Figure 10. A problematic shape-changing design.

Links overlap

Potential interference with path of light

Poor profile reproduction

Problematic dyad lengths Lack of symmetry

Figure 11. The preliminary selected shape-changing design.

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C. Analyzing/Modifying Mechanisms

The next step in the design process of shape-changing mechanisms is to analyze

the preliminary mechanism. This is accomplished using a basic physics simulation

product like the Working Model software package. With respect to shape-changing

rigid-body mechanisms, this design tool is capable of performing the following analyses:

1. Torque transmitted from the input motor.

2. Forces present at each revolute joint.

3. Positional accuracy.

The analysis of the preliminary design reveals a singularity right after the

mechanism forms the final design profile. This is due to certain links in the mechanism

reaching their maximum rotation as shown by the near-collinearity of the vectors [5] in

Figure 12. However, as the singularity does not occur during the critical portion of the

mechanism’s motion, it does not pose a problem. The major problem revealed during the

initial analysis is the relatively poor reproduction of the design profiles. These positional

inaccuracies are present in the Matlab representation of the design as well, but they are

particularly magnified after the design is translated into Working Model. Because of this

shortcoming, the design needed to be modified and re-analyzed.

Figure 12. Preliminary design near a singularity.

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The root of the problem was due to the inherent imbalance created by an

asymmetrical mechanism trying to represent a set of symmetrical design profiles. The

posed solution was to force the mechanism to be symmetrical through the manual

substitution and removal of links and joints. First, the center profile link was observed to

be undergoing translation and rotation. However, this link only requires translation in a

single direction to position itself correctly according to each design profile. The center

profile link only needs to move upward, however, its dyad provides for an excess of

motion and therefore adds unnecessary complexity to the mechanism. The solution is to

remove this dyad and replace it with a single prismatic joint that provides for only the

necessary motion. This substitution creates a zero degree of freedom system as shown by

Kutzbach-Gruebler’s equation [19] where M equals the degrees of freedom, L equals the

number of links, J1 equals the number of full joints, and J2 equals the number of half

joints.

3 1 2 3 11 1 2 15 0 0

However, it can be shown that the removal of one more dyad and its revolute joints

returns the me nchanism back to a si gle degree of freedom.

3 1 2 3 10 1 2 13 0 1

Therefore, in an effort to attain symmetry, one of the dyads of the four-bar sub-linkage

should be removed. This results in a mechanism with one prismatic link in the center,

and a set of two dyad connected profile links on either side of it. At this point in the

design, there is no longer any reason for the left and right side of the mechanism to be

dissimilar as they should be performing the same function. The mechanism on the left of

the prismatic link is mirrored to replace the mechanism on the right, forming a

completely symmetrical mechanism. The steps of this process are illustrated in Figure

13.

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Remove and replace with prismatic

Remove

21

Mirror a side to create symmetry 3 4

Finish

Figure 13. Steps taken to modify the preliminary design in order to create symmetry.

Based off the above procedure, it can be conjectured that a mechanism generated

by the ShapeChange program only needs to have one acceptable side according to the

criteria specified in Section B. That good side can then be mirrored according to the

above procedure. After completing a new iteration of this process with a new design

from ShapeChange, the mechanism in Figure 14 was derived. This mechanism exhibits

excellent surface accuracy as well as possesses 2 fixed revolute joints which share the

same location. This provides for facilitation in the prototyping stage. Also, the joints of

this mechanism do not undergo any considerable forces during the critical portion of its

stroke. However, there is a singularity present after the formation of the final design

profile, similar to the singularity of the original preliminary mechanism.

P a g e  | 22  

Figure 14. Kinematic blueprint for the final design.

D. Designing a Prototype Mechanism

The design of the first prototype of a given concept marks the initial attempt at

combining the theoretical with the practical. At this point in the design process of a

shape-change mechanism, a plausible blueprint has been developed for a parabolic light

reflector. However, in order to develop this concept into something that can actually be

built, several design challenges needed to be overcome. These design challenges were

approached knowing that resources (money and equipment) would be scarce. Therefore,

the number of off-the-shelf components was maximized and the amount of machined

parts was kept to a minimum. This resulted in a bulkier but more affordable prototype.

The first of these design challenges involved fabricating the profile links. This

required creating elongated curved links that are reflective on one side and possess a

means for pinning each link to each other in the correct location. These parts needed to

be fabricated based on available resources which consisted of standard machining

equipment i.e. a mill and lathe. The resulting solution for each profile link included a

narrow rib with a reflective sheet of aluminum formed and adhered to the top of it. The

top surface of each rib was machined to provide the correct curvature for proper

adherence of the corresponding aluminum sheet. Each sheet was formed by an

impromptu jig and by hand. The use of thin aluminum sheets to provide the reflective

surface area greatly reduced weight and waste material. The machined ribs also served as

a space to drill tapped holes in order to fasten other required hardware. This hardware

included miniature piano hinges to provide the revolute joints between each rib.

P a g e  | 23  

Miniature piano hinges were used because they would provide a robust but subtle means

of joining each profile link. Slots machined into each aluminum panel provide clearance

for the piano hinges. Also, attached to each rib were threaded rods with a spherical rod

end for coupling to a dyad at the proper location. The center link, however, instead had

rods attached for creating a prismatic joint. The profiles links are illustrated in Figure 15.

Figure 15. Outer profile link, inner profile link, center profile link.

Table 2. Parts included in Figure 15.

Part Number Description Part Number Description

1 Outer Reflective Panel 7 Inner Rib

2 Outer Rib 8 Inner Profile Rod

3 Piano Hinge 9 Center Reflective Panel

4 Outer Profile Rod 10 Center Rib

5 Spherical Rod End 11 Center Linking Rod

6 Inner Reflective Panel

The next design challenge involved fabricating the dyad links and connecting

them from the profile links to a fixed ground structure. The design of the dyads was

simplified as much as possible, consisting of only a threaded rod with a spherical rod end

connected to one end and a clevis rod end connected to the other. The spherical rod ends

of the dyads were coupled to the spherical rod ends from each profile link. Spherical rod

ends were used due to their ability to compensate for positional discrepancies due to

manufacturing inaccuracies. The clevis rod end of each dyad was then coupled to rods

attached to a frame creating a set of fixed revolute joints. This frame was also used to

P a g e  | 24  

correctly position a fluorescent tube at the fixed focus of the morphing parabolic panels

as well as provide a stable surface to position the sliding prismatic joint of the center

profile link. The dyad links and base assembly is illustrated in Figure 16.

Figure 16. Outer dyad link, inner dyad link, base assembly.

Table 3. Parts included in Figure 16.

Part Number Description Part Number Description

5 Spherical Rod End 15 Side Base Plate

12 Outer Dyad Rod 16 Center Base Rod

13 Clevis Rod End 17 Side Base Rod

14 Inner Dyad Rod 18 Center Base Plate

A considerable design challenge involved with translating the theoretical

blueprint for a shape-changing mechanism into a practical assembly was the issue of

interference. As shown in Figure 17, every dyad crosses every other dyad at some point

throughout the motion of the mechanism. The majority of the interference issues can be

resolved by staggering each dyad and profile link pair depth-wise (into the plane of the

paper). This also involves each rib being staggered onto a specific location respective to

its reflective panel which spans the entire depth of the mechanism. However,

interference issues still existed despite this solution. No link can travel past the fixed

revolute joint of the two inner dyad links due to the span of the rod holding the joints of

P a g e  | 25  

those dyads in space. Therefore, the threaded rods of the outer profile and dyad links

needed to contain an angle bend in order to avoid this interference.

Figure 17. Motion of the final design overlaid with the design profiles.

The final design described above was the result of several earlier design

iterations. Included in these iterations are the designs pictured in Figures 18. The first

figure shows the design of an asymmetrical mechanism. This iteration resulted in

unsatisfactory asymmetrical curves. The second figure is an attempt to create a shape-

changing paraboloid from only rigid bodies. This design faces several interference

issues. Some of these issues can be resolved by the introduction of compliance. The

P a g e  | 26  

third figure shows an earlier effort to attain symmetry through the design of a two

mirrored half mechanisms. The inherent problem of this design was the substantial

number of parts required. A large number of dyads were required including 2 four-bar

sub-linkages. Furthermore, additional mechanism would need to be added in order to

synchronize the 2 separate half mechanisms. The fourth figure depicts an alternate

version of the final design. This version includes an elongated depth to accommodate a

longer fluorescent tube as well as a second set of dyads to compensate for the additional

length. Each dyad and its counterpart would be tied together via threaded rods to

increase the system’s stability. Figure 19 shows the final design.

(1) (2)

(3) (4)

Figure 18. Previous design iterations.

P a g e  | 27  

Figure 19. Final design.

P a g e  | 28  

5. Resultant Prototype

The resulting prototype (Figure 20) functioned as predicted. It was able to

symmetrically form the desired design profiles, however, with limited accuracy. As a

result, the performance of the prototype fell short of expectations. Although an increase

in intensity was observable as evidenced in Figure 21, the ideal 43% increase in intensity

was not achieved due to manufacturing inaccuracies. Specifically, the improvised jig

used to bend the reflective panels did not do so in an effective manner. Therefore, some

kinked and irregular shaped panels were produced. This greatly hindered the collimation

effort. Furthermore, the miniature piano hinges provided less accuracy than was hoped.

In retrospect, flexural joints similar to those used in a lumped compliance mechanism [4]

may have been a better option. Lastly, the prismatic joint between the center profile

link’s rods and the fixed frame also allowed for too much error and too short of a sliding

contact surface. This issue could be resolved through the addition of linear bearings for

these rods.

Figure 20. Functional prototype.

P a g e  | 29  

Figure 21. Prototype at low intensity/high spread configuration on left. Prototype at high intensity/low spread configuration on right.

6. Conclusion

Rigid-body mechanisms offer a morphing capability to a wide array of

applications. In the author’s opinion, the most promising applications presented in this

thesis are the ergonomic chair and the light reflector. The prototype of the light reflector

proved to be a success, but not without a fair share of shortcomings related to fabricating

a first prototype. Nonetheless, if the final design presented in this thesis were redesigned

for manufacture and assembly, it may prove to be viable for a commercial task lamp

application.

As the development of rigid-body shape-changing mechanisms continues, it is

recommended that specific attention be given to the integration of compliant skin

materials with load-bearing rigid-body structures. This technique utilizes the inherent

advantageous of both materials. Current research efforts in this area include the shape-

changing airfoil work being performed by Cornerstone Research Group [6] and the

shape-changing automobile work being performed by BMW [20]. It is proposed that

these and similar research efforts can greatly benefit from the kinematic synthesis

described in Chapter 4 of this thesis.

P a g e  | 30  

References

[1] Weisshaar, T.A. (2006), Morphing Aircraft Technology – New Shapes for

Aircraft Design, Multifunctional Structures / Integration of Sensors and Antennas,

pp.O1-1 – O1-20.

[2] Gano, S.E., Perez, V.M., Renaud, J.E., Batill, S.M., and Sanders, B. (2004),

Multilevel Variable Fidelity Optimization of a Morphing Unmanned Aerial Vehicle,

Collection of Technical Papers – AIAA/ASME/ASCE/AHS/ASC Structures, Structural

Dynamics and Materials Conference, vol. 4, pp 2777-2792.

[3] Feldman, H.A. (2007), Space-based Antenna Morphing Using Reinforcement

Learning, Collection of Technical Papers – AIAA Aerospace Sciences Meeting, vol. 3, pp

1978-1986.

[4] Ananthasuresh, G.K., and Kota, S. (1995), Designing Compliant Mechanisms,

Mechanical Engineering, vol. 117, no. 11, pp. 93-96.

[5] Murray, A.P., Schmiedeler, J.P., and Korte, B.M. (2008), Kinematic Synthesis of

Planar, Shape-Changing Rigid-Body Mechanisms, Journal of Mechanical Design,

Transactions of the ASME, vol. 130, no. 3.

[6] Hermiller, J.M., Cable, K.M., Hemmelgarn, C.D., Qi, H.J., and Castro, F. (2009),

Thermal Design Methodology for Attaching Morphing Components, Proceedings of

SPIE – The International Society for Optical Engineering, vol. 7290.

[7] Kaleta, J., Bomba, J.M., Lewandowski, D., and Wiewiorski, P. (2006), Smart

Magnetic Materials and Magnetoresistive Sensors in Controlling of Mechanical

Structures, Proceedings of SPIE – The International Society for Optical Engineering, vol.

6167.

P a g e  | 31  

[8] Li, M., Chen, W., and Jia, L. (2009), Application of Piezoelectric Actuators to

Aircraft Aeroelastic Performance Enhancement, Hangkong Xuebao/Acta Aeronautica et

Astronautica Sinica, vol. 30, no. 12, pp. 2301-2310.

[9] Nagata, M., and Yamamoto, Y. (2009), Synthesis and Characterization of

Photocrosslinked Poly(ε-caprolactone)s Showing Shape-Memory Properties, Journal of

Polymer Science, Part A: Polymer Chemistry, vol. 47, no. 9, pp. 2422-2433.

[10] Baier, H., Datashvili, L., and Rapp, S. (2009), Enhancing Space Satellite

Performance by Integrating Smart Sensors and Actuators for Sensing and Shape

Morphing, Proceedings of SPIE – The International Society for Optical Engineering, vol.

7493.

[11] Chandak, A., and Somani, S.K. (2009), Design of Multistage Evaporators for

Integrating with Scheffler Solar Concentrators for Food Processing Applications,

International Solar Food Processing Conference.

[12] Mancini, T., Heller, P., Butler, B., Osborn, B., Schiel, W., Goldberg, V., Buck, R.,

Diver, R., Andraka, C., and Moreno, J. (2003), Dish-Stirling Systems: An Overview of

Development and Status, Journal of Solar Energy Engineering, Transactions of the

ASME, vol. 125, no. 2, pp. 135-151.

[13] Loss of the Lumber Curve in the Driving Seat: A Twenty Person Study (1996),

British Osteopathic Journal, vol. 19, pp. 19-23.

[14] Porter, J.M., and Norris, B.J. (1987), The Effects of Posture and Seat Design on

Lumbar Lordosis. Human Factors in Transport Design, Contemporary Ergonomics, pp.

191-196.

P a g e  | 32  

[15] Bangtsson, E., Noreland, D., and Berggren, M. (2003), Shape Optimization of an

Acoustic Horn, Computer Methods in Applied Mechanics and Engineering, vol. 192, no.

11-12, pp. 1533-1571.

[16] Fukuda, H., Yanagimoto, K., China, H., and Nakagawa, K. (1995), Improvement

of Vehicle Aerodynamics by Wake Control, JSAE Review, vol. 16, no. 2, pp. 151-155.

[17] Akashi, Y., and Boyce, P.R. (2006), A Field Study of Illuminance Reduction,

Energy and Buildings, vol. 38, no. 6, pp. 588-599.

[18] Decker, D. (2006), Adaptive Headlights Aim to Ease Nighttime Driving Hazards,

Photonics Spectra, vol. 40, no. 12, pp. 54-58.

[19] Norton, R.L. (2008), Design of Machinery: An Introduction to the Synthesis and

Analysis of Mechanisms and Machines, Fourth Edition, McGraw Hill.

[20] The BMW GINA Light Visionary Model. Innovative Approach and Optical

Expression of Creative Freedom (2008), BMW Media Information.

P a g e  | 33  

Appendix A: Engineering Drawings

P a g e  | 34  

Appendix A: Engineering Drawings continued

P a g e  | 35  

Appendix A: Engineering Drawings continued

P a g e  | 36  

Appendix A: Engineering Drawings continued

P a g e  | 37  

Appendix A: Engineering Drawings continued

P a g e  | 38  

Appendix A: Engineering Drawings continued

P a g e  | 39  

Appendix A: Engineering Drawings continued

P a g e  | 40  

Appendix A: Engineering Drawings continued

P a g e  | 41  

Appendix A: Engineering Drawings continued

P a g e  | 42  

Appendix A: Engineering Drawings continued

P a g e  | 43  

Appendix B: Purchased Parts

Qua

ntity 

8  4  2  1  1  1  1  1  2  1  1  1  2  16 

1  1 

Ven

dor 

Midwest C

ontrol Prodcuts Co

rp. 

Midwest C

ontrol Prodcuts Co

rp. 

Light B

ulb Dep

ot 

Light B

ulb Dep

ot 

Light B

ulb Dep

ot 

Light B

ulb Dep

ot 

Light B

ulb Dep

ot 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

Description

 

10‐32, Aluminum

, Nylon

 race, Fem

ale 

10‐32, Aluminum

 

T5, 5.83", 410

0 Ke

lvin 

Mini Bipin 

120 V 

With

 con

denser 

600 V max, 660

 W m

ax 

8‐32

, Spade

 head, 18‐8 Stainless 

Piano hinge, Stainless 

24"x12

", M

irror finish, Alloy 50

52 

2‐56

 flathe

ad, 1/8", Stainless 

2‐56

 flathe

ad, 3/16", Stainless 

8‐32

 hex, 1", Aluminum

 (blue) 

Unthreade

d, Aluminum

, Screw

 size 4 

4‐40

, Aluminum

, 1" length 

4‐40

, Aluminum

 

Item

 

Sphe

rical Rod

 End

 

Clevis Rod

 End

 

Fluo

rescen

t Bulb 

T5 Socket 

Ballast 

Starter 

Starter Ba

se 

Thum

b Screw 

Hinge 

Aluminum

 She

et 

Machine

 screw

s 

Machine

 screw

s 

Socket cap

 screw

s 

Spacer 

Threaded

 Rod

 

Nut 

Prod

uct No. 

APF‐3 

BTCA

‐187

S 

0053

9B 

2905

3A 

2651

6A 

0094

0A 

0093

0A 

9174

4A19

7 

1155

5A1 

8202

K31 

9308

5A01

0 

9308

5A01

5 

9851

1A45

3 

9251

0A02

0 

9322

5A47

1 

9318

1A00

3 

P a g e  | 44  

Appendix B: Purchased Parts continued

Qua

ntity 

1  1  1  2  1  1  1  1  1  1  2  1  1  1 

Ven

dor 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

McM

aster‐Ca

rr 

Description

 

10‐32, Aluminum

 

Coun

tersun

k external te

eth, #10

 

Aluminum

, #4 

3M DP4

20, B

lack, 37 mL, 2:1 Duo

‐Pak 

2:1 Plun

ger for App

licator Gun

 

For 3M

 Duo

‐Pak cartridges 

Horsehair brush, steel handle 

10‐32, Stainless, 24" length 

10‐32, Aluminum

, 36" length 

Ø 1/4", Aluminum

, 36" length 

1/4", A

luminum

, One

‐piece clamp‐on

 

Aluminum

, #8 

Plastic, #4 

Stainless, #4 

Item

 

Nut 

Locking Washe

r 

Washe

r 

Epoxy Adh

esive 

Plun

ger 

Mixer Nozzle 

App

licator brush 

Threaded

 Rod

 

Threaded

 Rod

 

Rod 

Shaft collar 

Washe

r 

Retaining Washe

r 

Locking Washe

r 

Prod

uct No. 

9318

1A41

1 

9090

0A01

2 

9328

6A00

5 

7467

A51

 

7467

A48

 

7467

A12

 

7237

T82 

9880

5A01

1 

9443

5A33

6 

9062

K263

 

6157

K12 

9328

6A00

9 

9175

5A20

5 

9844

9A51

0