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Compliant pediatric prosthetic knee
h [electronic resource] /
by Sebastian Mahler.
[Tampa, Fla.] :
b University of South Florida,
ABSTRACT: We have designed and examined a compliant knee mechanism that may offer solutions to problems that exist for infants and toddlers who are just learning to walk. Pediatric prosthetic knees on the market today are not well designed for infants and toddlers for various reasons. Children at this age need a prosthetic that is light in weight, durable, and stable during stance. Of the eleven knees on the market for children, all but three are polycentric or four-bar knees, meaning they have multiple points of movement. Polycentric knees are popular designs because they offer the added benefit of stable stance control and increased toe clearance, unfortunately this type of knee is often too heavy for young children to wear comfortably and is not well suited for harsh environments such as sand or water, common places children like to play. The remaining three knees do not offer a stance control feature and are equally vulnerable to harsh environments due to ball bearing hinges.Compliant mechanisms offer several design advantages that may make them suitable in pediatric prosthetic knees -- light weight, less susceptible to harsh environments, polycentric capable, low part count, etc. Unfortunately, they present new challenges that must be dealt with individually. For example compliant mechanisms are typically not well suited in applications that need adjustability. This problem was solved by mixing compliant mechanism design with traditional mechanism design methods. This paper presents a preliminary design concept for a compliant pediatric prosthetic knee. The carbon fiber composite spring steel design was first built and then evaluated using Finite Element Analysis. The prototype's instant center was plotted using the graphical method. From our analysis position, force and stress information was gathered for a deflection up to 120 degrees. The instant centers that were plotted indicate that the knee has good potential in offering adequate stability during stance.
Thesis (M.S.)--University of South Florida, 2007.
Includes bibliographical references.
Text (Electronic thesis) in PDF format.
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
Title from PDF of title page.
Document formatted into pages; contains 79 pages.
Advisor: Craig Lusk, Ph.D.
Pseudo rigid body model.
Finite element analysis.
Above knee amputation.
x Mechanical Engineering
t USF Electronic Theses and Dissertations.
Compliant Pediatric Prosthetic Knee by Sebastian Mahler A thesis submitted in partial fulfillment of the requirement s for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Craig Lusk, Ph.D. Rajiv Dubey, Ph.D. Nathan Crane, Ph.D. Date of Approval: November 5, 2007 Keywords: compliant mechanism, pseudo rigi d body model, instant center, finite element analysis, 4-bar, func tional requirements, pediatri c prosthetic, prosthetic, above knee amputation, transfemoral Copyright 2007 Sebastian Dario Mahler
i Table of Contents List of Figures iii Abstract v Chapter One Introduction 1 Motivation 2 Contributions 5 Chapter Two Background Phases of Gait 6 Kinematics of the A natomical Knee 10 Instantaneous Center 11 Polycentric Knees 13 Issues Facing Pediatric Prosthetic Users 14 Design Criteria 16 Compliant Mechanisms 17 Conclusions 24 Chapter Three Adjustable Comp liant Pediatric Knee 25 Chapter Four Position/Displacement Analysis 29 Conclusion 35 Chapter Five Force and Stress Analysis 37
ii Conclusion 45 Chapter Six Discussion and Conclusions 47 References 49 Appendices 51 Appendix A Ansys Code Used to Find Force and Stress Data 52
iii List of Figures Figure 1.1 Adjustable compliant pediatric knee prototype 1 Figure 2.1 The five phases of stance 7 Figure 2.2 Instantaneous c enter of rotation of a rolling wheel 12 Figure 2.3 MightyMite 4-bar knee by Fillauer 14 Figure 2.4 Scissor jack mechanism used to increase mechanical advantage 18 Figure 2.5 Plastic bottle cap with compliant hinge 19 Figure 2.6 Fully compliant injection molded plastic box 20 Figure 2.7 Pencil box with movable metal hinges 21 Figure 4.1 Rotational plots using finite element analysis, showing the motion of points F and C when the shank is forced to rotate relative to the socket having a bracket angle for = 64 30 Figure 4.2 Instantaneous center of rota tion path of the compliant knee 32 Figure 4.3 Instant cent er path found in a fou r-bar knee design 33 Figure 5.1 Prototy pe design labeled 39 Figure 5.2 Paths used to generate force plots measured in meters 39 Figure 5.3 Force plot data of t he compliant knee in kg on a log10 scale utilizing overlapping of multiple data sets with a bracket angle of 50 degrees 40 Figure 5.4 Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 80 degrees measured in meters 40
iv Figure 5.5 Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 70 degrees measured in meters 41 Figure 5.6 Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 60 degrees measured in meters 41 Figure 5.7 Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 50 degrees measured in meters 41 Figure 5.8 Stress plot data of t he compliant knee in GPa on a log10 scale with a bracket angle of 80 degrees measured in meters 43 Figure 5.9 Stress plot data of t he compliant knee in GPa on a log10 scale with a bracket angle of 70 degrees measured in meters 43 Figure 5.10 Stress plot data of the compliant knee in GPa on a log10 scale with a bracket angle of 60 degrees measured in meters 44 Figure 5.11 Stress plot data of the compliant knee in GPa on a log10 scale with a bracket angle of 50 degrees measured in meters 44 Figure 5.12 Corresponding plot color for 1.758 GPa on a log10 scale 45
v Compliant Pediatric Prosthetic Knee Sebastian Mahler ABSTRACT We have designed and examined a comp liant knee mechanism that may offer solutions to problems that exist for infant s and toddlers who are just learning to walk. Pediatric prosthetic knees on t he market today are not well designed for infants and toddlers for various reasons. Children at this age need a prosthetic that is light in weight, durable, and stabl e during stance. Of the eleven knees on the market for children, all but three ar e polycentric or four-bar knees, meaning they have multiple points of movemen t. Polycentric knees are popular designs because they offer the added benefit of stable stance control and increased toe clearance, unfortunately this type of knee is often too he avy for young children to wear comfortably and is not well suited for harsh environments such as sand or water, common places children like to play. The remaining three knees do not offer a stance control feature and are equa lly vulnerable to harsh environments due to ball bearing hinges. Compliant mechanisms offer several design advantages that may make t hem suitable in pediatric pr osthetic knees Â– light weight, less susceptible to harsh env ironments, polycentric capable, low part count, etc. Unfortunately, they present new challenges that must be dealt with individually. For example compliant mec hanisms are typically not well suited in
vi applications that need adjustability. This problem was solved by mixing compliant mechanism design with traditi onal mechanism design methods. This paper presents a preliminary design concept for a compliant pediatric prosthetic knee. The carbon fiber composite spri ng steel design was first built and then evaluated using Finite Element Analysis. The prototypeÂ’s instant center was plotted using the graphical method. From our analysis position, force and stress information was gathered for a deflection up to 120. The instant centers that were plotted indicate that the knee has good potential in offering adequate stability during stance.
1 Chapter One Introduction The objective of my research is to desi gn an adjustable compliant pediatric knee. This knee will better suit children by loweri ng the over-all weight of the knee while keeping the benefits of heavier rigid link knees. An adjustable knee design is described and its position, force, and stress properties are analyzed. Figure 1.1. Adjustable compliant pediatric knee prototype
2 Motivation Our goal is to improve current pediatri c knee technology by designing a knee having lower weight, reduced cost, and increased functionality. By using a compliant-mechanism-based design we are capable of potentially lowering the weight of the knee substantially, and in doing so we will allow younger children to more comfortably wear their prosthetic Some pediatric knees on the market today offer great stability as well as some of the benefits found in adult prosthetic knees but they are typically too heavy fo r young children to wear for long periods of time. The benefit of having a child wear t heir prosthetic longer is that it allows them more time to adapt to wearing their device. By giving a child time to adapt to wearing a prosthetic prior to walkin g you promote the child to walk at a younger age. Standing and walking up-right pr omotes the childÂ’s ability to move about with their hands free to grasp objects and inte ract with new surroundings that were otherwise unattainable by cra wling. This interaction with their new found world stimulates positive m ental and physical growth.  To lower the weight of the overall pr osthetic limb, a peg leg is typically prescribed to give the child a way of walking that is easy to use but this oftentimes results bad habits acquired from its use which are difficult to rid. The peg type leg requires that it be adjusted to a length shorter than the sound limb to make it useful. This misalignment forces the child to walk as if one foot was constantly in a hole. This creates problems for the childÂ’s gait, forcing the hips to excessively deviate from normal gait to compensate for the difference in height
3 between the two limbs. This constant misa lignment is not only bad for the body, it also produces habits which make adapti ng to better technology later in life become more difficult. By instilling good habits from an articulated knee, like proper hip alignment and good we ight transfer, you assist the child now as well as later in life. A better knee design could potentially o ffer a reduction in shock that is associated with many prosthetic knees. Because our design is compliant, i.e. allows for elastic deflection, shock felt from ground reaction forces can be greatly reduced. Another benefit our design is t he continuous connection that is made with the lower leg and is created by a llowing torque to be transmitted through the knee. Prosthetic knees are typically unabl e to transmit torque through their ball bearing hinges. This feeling through t he knee allows the patient to have an increased confidence as to where their lo wer leg is located during gait, reducing the fear and potential of falling. Children are rough on their bodies and es pecially rough on prosthetics. Children are notorious for playing in plac es that most prosthetics would not be able to endure. For this reason we intend to design a knee that would be particularly suited to children. The co mpliant knee would be made impervious to sand, water, and other corrosive agents. By eliminating ball be arings or other sensitive equipment, the knee can be s ubmerged in liquid or exposed to sand without causing serious damage to its func tion. Our design could potentially out last other designs on the market simply because it does not have limitations as to
4 where it can be used. Because children have a mind of their own it is tough to keep them away from potential hazards to their prosthetic. Given the potential issues that child ren present to prosthetic devices, small alterations to the design could pot entially create a produc t that would out perform current technology. The compli ant knee offers more function and versatility than the competition by offeri ng a lighter weight design with additional features only found in heavier adult pr osthetics. Our design offers shock absorption as well as a cont inuous connection to the prostheticÂ’s lower limb. The knee would be impervious to environmental hazards making it a better choice for small children. For these reasons we feel compliant knees could potentially take over the market for small children.
5 Contributions Design and examined an adjustable pediatric compliant knee prototype Created a prototy pe weighing less than five ounces. Formulated a method for analyzing the rotation and translational motion of the compliant knee using nonlinear FEA accomplished through calculation and plotting of the instant cent ers of the kneeÂ’s rotation Developed a method for simultaneously calculating the external reaction forces and internal stresses of the knee for the anticipated region of motion for a deflection of for 0< <120.
6 Chapter Two Background Phases of Gait Wheeless' Textbook of Or thopedics (2000)  describes the phases of gait in two different sections, the stanc e phase and the swing phase. In stance phase, the foot is purposefully in cont act with the ground, while in swing phase, the foot is suspended in air. The stance phase compri ses the larger segm ent of the entire gait cycle, approximately 60 per cent of the cycle, and can be broken down into five separate parts . When an individual demonstrates normal gait, there is a level of symmetry to the process and t here is a consistency to the balance between the 60 percent stance phase and 40 percent swing phase . When variations in the ratio of stance phas e to swing phase exists, abnormalities can be noted in the gait. The stance phase, illustrated in Figure 2.1 includes initial contact (made by the heel of the foot fo llowing an initial upswing), loading response, in which the foot balance is secured and alignm ent occurs between the hip, knee and ankle to place weight on the foot, midstance (entirety of the body weight is on
7 the foot, knee is extended and the ankle is neutral), terminal stance (the end stage and initial lifting), and toe off, wh ich ends at the toe lift . Figure 2.1. The five phases of stance The key to the stance phase is resist ance to falling. Stability and the alignment of the trunk over the base of support which in normal cases, consist of the alignment of hip, knee and ankle ov er the foot are imperative to the progression from the stance phase to the swing phase. Stability refers to the balancing of the center of mass over the base of support and also resistance to knee buckling, both of which often resu lt in collapse of the amputee. Knee buckling is often caused by the flexion moment, illustrated in Figure 2.1, which is present during all phases of stance with t he exception of terminal stance. This exception is caused by the ground reaction fo rce falling anterior to the knee joint. Initial Contact Mid-Stance Loading Response Toe Off Terminal Stance
8 The swing phase is especially im portant when understanding the dynamics of a prosthetic knee and the func tionality. The swing phase comprises 40 percent of the total gait process, and begi ns with the toe lift-off . After the stance phase, the toe lifts of f the ground behind t he body. The leg then begins to move forward and the foot extends, in alignment with knee flexion. Subsequently, the forward motion of the body allows for the swing of the leg forward. Key to this, though, is the abi lity of the toe to clear the ground. In unbalanced prosthetics, it is not uncomm on for individuals to purposefully use pelvic rotation to compensate for the inabilit y to move the foot or clear the toe, and so prosthetics that are improperly fitt ed can result in major gait complications that arise from attempts to adjust for the swing phase . The concept of prosthetic gait synergy relates the functionality of the prosthetic to the capacity to mainta in normal gait through motor control. Essentially, the prosthetic has to work in correlation with existing body structures to allow motor control, create repeatable pa tterns of muscular activity and work in concert with body kinetics . Movements that are syner gistic actually require minimal immediate thought processe s or neural control and so have been identified as "automatic" in their performance . One of the keys to prosthetic development is the ability to integr ate synergistic movements and support independent control of gait in or der to create balanced movement. Prosthetic gait synergy, then, is "t he best possible gait pattern of an amputee with a given type of prosthesis/p rostheses. Not every deviation from
9 normal gait is considered a component of the prosthetic gait synergy. Only deviations that remain apparent after pr oper residuum conditioning, socket fit, and prosthesis alignment and adjustmen t, and after the amputee becomes accustomed to the prosthesis" . Dundass, Zao and Mechefske (2003) studi ed the use of a hydraulic knee controller in a transfemoral amputee subject, and identified prosthetic deterioration as a significant issue for transfemoral amputees. This study is especially significant for addressing the issue of pediatric transfemoral amputees because it is likely that deterioration will occur because of the longevity of prosthetic use and the need to identify the impacts of growth from the transfemoral amputee. Dundass, Zao and Mechefske (2003) i dentify kinetic gait analysis as one of the significant measures of the inte rnal and external fa ctors that influence motion. These researchers maintain that there are some fact ors that influence the outcomes of assessment kinetic gait proce ss, the least of which is that stride differentials and the fact that many am putees do not put thei r full body weight on their prosthetic limb during the stance phas e, suggests that these are variables that should be considered when creat ing prosthetic devices . What is evident is that the majo r purpose of the prosthetic devices, including prosthetic knee components, is the effort to re create human knee and leg function and the synergy betw een physiological components and the prosthetic utilized. In doing this, it is valuable to consider the kinematics of the
10 knee and the influence of the knee struct ure on the development of prosthetic prototypes. Kinematics of the Anatomical Knee The structure of the knee is not unlike the structure of the elbow, where three distinct bones come to a central point. The femur, tibia and fibula are static and strong support mechanisms for the musculatu re and the load forces that result from movement like taking a step are Â“transmitted over the hyaline joint cartilage, allowing a smooth and easy motion of the joint surfaceÂ” . Â“The femorotibial joint surfaces are noticeably incongruent and offer just a two-point contact area. The hip joint, in contrast with its s pherical head in its socket, as well as the ankle, have large c ontact surfaces for l oad transferÂ” . This clear differentiation between the stru ctural elements of the lower extremity raises the question of how the effect ive link between the skeletal components and the musculature is reached and how the formation of the central pivot of the knee is realized . Essential to the formati on of the knee itself are two cruciate ligaments and they assist in the creation of a mechani cal system of Â“crossed four-bar linkageÂ” which allows the knee to mo ve in Â“three rotations (ext ension-flexion, external
11 rotation-internal rotation, and varus-val gus rotation) and in three translations (anteroposterior, mediolateral, and co mpression-distraction)Â” . When prosthetics function in the place of the st ructure of the knee, it is necessary to develop systems that mimic the ro tational component of the knee. Because of the necessity for flexibility and a range of motion, the ligaments, musculature and support compon ents of the knee work in concert to allow for both rotation and stabilization. The prosthetic must represent the capacity for load transmission from the fe mur to the tibia (o r the structural equivalents in the prosthetic ). This must include an id entification of methods to absorb or reduce the peak load forces and their impacts on the structures surrounding the joint. Instantaneous Center The instantaneous center (IC) in a plane or in a plane figure which has motions both of translation and of rotation in the plane is the point which for the instant is at rest. (WebsterÂ’s 1913) A two dimensiona l instant center is a location within a plane that a body instantaneously rotates around. At this instant all motion within the body is traveling perpendicular to the line that connects the point of interest on the body to the IC. Figure 2.2 illustrates how the instant center of the rotation of a wheel rotating on flat plane is always located at the point of contact
12 with the surface. Notice how the motion of points A and B are traveling perpendicular to the line that intersects t he instant center. This is the case because at any instant all locations on a rotating body travel perpendicular to the IC. Figure 2.2. Instantaneou s center of rotation of a rolling wheel ICs are calculated in order to fully understand motion characteristics when designing mechanisms. This is especia lly true for mechanisms designed to be used as prosthetic knees. The IC of a prosthetic knee determines how stable the knee will be during stance phase. The proper IC location is capable of assisting in toe clearance as well as provide vita l control feedback, i.e. gait synergy, to patients with short residual limb s. i.e. if the IC is placed near the end of the residuum the amputee is afforded greater c ontrol over the prosthesis. For these reasons it is important to understand w here the ICs are loca ted during various angles of flexion. The IC for a single-ax is knee is located at the center of the
13 joint and doesnÂ’t change during rotation. For more complicated, polycentric knees, knees without a fixed center of rotati on, this is not the case. During their full range of motion, the IC undergoes constant change. Polycentric Knees There are currently more than 100 differ ent types of prost hetic knees on the market today, eleven of which are pediatri c knees. Of the many different types available there are two main types of cat egories that they can fall into, single axis and polycentric. Single-axis knees are designed with one pin joint located near the position of the anatomical knee. F our and six-bar knees fall under another category known as polycentric and have i rregular patterns which the ICs follow. Polycentric knees, like the Mighty Mite illustrated in Figure 2.4, have traits that are beneficial for a various reasons. Polycentric knees increase toe clearance during swing phase, they decrease the length of the upper-leg and appearance of abnormality during sitting. Polycentric k nees are also particularly stable during stance phase.
14 Figure 2.3. MightyMite 4-bar knee by Fillauer Issues Facing Pediatric Prosthetic Users Wilk et al. (1999) noted t hat the common practice wit h very young children with amputations above the knee is to fit the ch ild with a prosthetic of some kind as soon as possible, as soon as the child is able to pull to a standing position, which generally occurs between the ages of 9 and 16 months. It is common to begin gait training with very young amputees ut ilizing non-articulating prosthetics, i.e. no knee joint present, because young chil dren can learn to walk without a functional knee easier with the non-ar ticulating prosthetics, and because commercially available pediatric prosthetic devices with articulating knees are not generally made available . The key consideration here is that when the transition to an articulating knee is m ade, there is often a longer period of adjustment . In addition, researcher s have noted that knee prosthetics have
15 relatively low durability and that young ch ildren who are so active may require multiple prosthetics that can be cost prohibitive. Researchers have recognized, though, t hat whether cost-prohibitive or necessary, an increasing focus on the past processes and the negative implications for pediatric patients shoul d be noted. It is not uncommon, for example, for young children to be fitted with prosthetics without articulating knees and then go many years before a new prosthetic is prov ided. Children in these situations often incorporate different met hods of gait control that are difficult to change after years of use. As a resul t, the benefits of in tegrating improved systems can be demonstrated through the us e of articulating knee prosthetics from the onset. The result of the use of non-articu lating knee prosthetics for young children is that they often result in gait variations that can be difficult to address when fitting them for differ ent prosthetics later in lif e. Wilk et al. (1999) maintained though, that children who had prosthetics without articulating knees prior to the use of articulated system s demonstrated immediat e adaptations in their gate, and improvements in hip flexion-extensio n, unbalanced pelvic motion and improvements in balance. Wilk et al. (1999) argued that children can be fitted prior to 18 months of age with articulated knee prosthetics and t hat these prosthetics can reduce the chance of problematic gait deviations that cannot be corrected and improve the chances that pediatric patients will demonstr ate normal childhood activities. As a
16 result, these authors argue in favor of the development of im proved prosthetic devices for young amputees and the discardi ng of the old practice of the use of non-articulated prosthetics in very young children . Design Criteria In the paper written by Andrysek Â“Design Ch aracteristics of Pediatric Prosthetic KneesÂ”, functional requirements are examined within the pediatric user community. The main functional require ments are comfort, fatigue, stability, and resistance to falling. Sitting appearance and adequate knee flexion are of lower importance. Stance control and toe clearance were of the highest importance. These criteria were used to rate five knees on each of the five children in the study. From the study it is noted that a single-axis knee is equally suitable in meeting the highly and aver agely important functional requirements, meaning it is not as important for a child to have all of the benefits of technology, for proper gait as long as the importanc e is on function and not on appearance. The article explains that 8 of 11 comme rcially available knees are fouror six-bar configurations. They are said to be highly acceptable due to there ability to control stance phase, increased toe cl earance, and offer a more natural knee location. The disadvantages are that they are heavier than the single axis counterparts. The added technology hinders rather than helps especially with
17 small children due to the added weight. Th e results of the study suggest that a single axis knee with particular axis pl acement and a stance control mechanism satisfy the design parameters similarly to polycentric knee joints. The study was set up to design a more effective, and le ss complex knee to minimize the size and weight for the pediatric needs. The study concluded that a single axis knee has demonstrated the required functi on while fulfilling the functional requirements, which could result in a hi ghly functional, less complex, knee joint.  Compliant Mechanisms It is possible that the benefits of polycentric k nees can be achieved without prohibitive weight problems by using a design technology known as Â“compliant mechanisms.Â” A mechanism is a device that is intended to transfer motion, force or energy. Mechanisms are typically ri gid body and are made up of rigid links connected through movable joints. Me chanisms are common in nearly every part of our lives from a pair of scissor s to the steering syst em on your car. Mechanisms can be used to increase or decreases mechanical advantage depending on the purpose of its use. The scissor jack, shown in Figure 2.5, is an example of a mechanism that increases mechanical advantage and allows the user to lift objects much heavier than would be possible wit hout it. Although
18 mechanisms are capable of increasing output velocity or forces with respect to the input velocity and force, they are not capable of in creasing the energy output of the system. For example when the out put velocity is higher than the input velocity the output force must be lower than the input force and when the output force is higher than the input force the opposit e is true. This ability to transfer energy is not entirely efficient though. Energy is wasted by the mechanism due to losses in the system typically caused by friction. These losses can be small or large depending on the type and des ign of the mechanism. Figure 2.4. Scissor jack mechanism used to increase mechanical advantage A compliant mechanism is designed to do the same basic task as a traditional rigid link mechani sm, transferring motion, force, or energy, but it does
19 so by allowing deflection of one or mo re flexible members instead of using movable joints. One example of a co mpliant mechanism is the plastic hinge found on various plastic bottle caps shown in Figure 2.6 The motion of the cap is similar to that of a single axis hinge wit h the addition of bi-stable positioning. In other words the cap favors two distinct pos itions along its path of rotation. The advantage of this is that the bi-stabili ty keeps the cap open and out of the way while pouring the contents from the bottle. The ability to fall in either of two separate positions is achieved by the kinematics of the system in combination with the potential energy stored wit hin the flexible segments. Figure 2.5. Plastic bottle cap with compliant hinge
20 There are many reasons why compliant mechanisms have advantages over traditional rigid link mechanisms. Compliant mechanisms have the ability to greatly reduce the total number of parts and assembly steps needed to create a mechanism. These compliant mechanism s are oftentimes created from one injection-molded piece. An example of this type of fully compliant mechanism is a plastic injection molded box shown on in Figure 2.7 A similar design, shown in Figure 2.8, uses traditional moving hinges. No tice how the compliant box uses only one piece of material for the entire mechanism. This single piece mechanism fulfills the same task required from a pensile box but does not require any assembly after the initial molding pr ocess. The reduction in required parts and assembly steps for production can greatly reduce the costs and time spent fabricating mechanisms. Figure 2.6. Fully compliant injection molded plastic box
21 Figure 2.7. Pencil box with movable metal hinges Another key feature of compliant mechanisms is the reduction in movable joints. This greatly reduces the amount of internal motion within the mechanism which can potentially reduce the wear, friction, and need to lubricate parts. These attributes can be favorable in situations where the environment is corrosive or harsh. By lowering the amount of wear in a mechanism it is possible to extend the life of the device dramatically. The reduction in total friction losses can be favorable especially in the case of the pediatric knee. The reduction in friction losses can lower the amount of energy ex pelled by the child during its use which could extend the amount of time a child can comfortably wear the prosthetic. The ability of compliant mechanisms to operat e without lubrication can be favorable
22 for extending wear capabilities as well as removing the need to service the device for this cause. In addition to t hese benefits a reducti on in vibration and noise is possible due to the lower fricti on and absence of additional moving parts. Compliant mechanisms utilize flexing of segments to generate motion and in this flexing they store potential energy. This potential energy storage can easily be used as a built in feature of the mechanism. By storing energy in the form of strain energy a compliant mechanism can generate force feedback similar to externally mounted springs without the need to add additional parts. A good example of this is a bow and arrow syste m. The potential energy is initially put into the system by the pulling from the archerÂ’s arms. This potential energy within the bow is later released and tur ned into kinetic energy in the arrow once the bow is fired. Another feature of compliant mechanism s is the reduction in overall weight of the device. Rigid-body mechanisms r equire that the lin ks within the system remain rigid and therefore require higher strengths and additio nal material to build. By allowing the mechanism to flex under pressure, you lower the need for heavier rigid structures. This feature is especially beneficial to the project at hand. By lowering the weight of the pediat ric knee you allow younger children the ability to comfortably wear a prosthetic as well as increase the total time one can be worn by any patient. Compliant mechanisms are favorable in many situations that we may not naturally consider them for. The desire to use rigid link mechanisms is difficult to
23 get around due to the percepti on that strength and rigidity go hand in hand. This is not always true, especially in the ca se of compliant mechanisms. Compliant designs can offer many advantages over rigid type mechanisms and for these reasons we feel a compliant knee coul d easily be an improvem ent over current technology. Unfortunately compliant mechanisms are not void of weaknesses. Compliant hinges are not well suited and c ould potentially be subject to failure under these conditions. Excessive flexural deflection Cyclic fatigue Compressive flexural loading Specific design consideration must be made to prevent such failure. Limiting of the maximum deflection can be accomplished with pr oper design. By limiting the flexural deflection, fatigue associated with cyclic loading is greatly reduced. Proper location of the compli ant hinges during desi gn is required to prevent compressive failure. 
24 Conclusions It has been readily recognized that the corr ect alignment of the prostheses is imperative not only to successf ul mobility, but also to the longevity of prosthetic use. Longevity and durability are determined by a number of factors, including the underlying causative factor for prosthetic use and the comfort and control that are present as a result of the prosthetic. Over the course of the last two decades, considerable changes in the development of prosthetics for transf emoral amputees have developed. It has been recognized that the development of mo dern prostheses reflects the focus on this type of amputation and the need for young children to have access to functional prosthetics from an early age. Good prosthetic design requires toe cl earance, stability, light in weight, and adjustability. To classify knees FEA a nd calculation of the instant center of rotation can help in determining toe clear ance and stability. Use of compliant mechanisms should reduce the weight of the prosthetic and allow young children the benefit of an articula ting prosthetic knee. Evidence suggests that there is a cons iderable level of improvement that can be made in the existing prosthes es for pediatric patients, including improvements that will continue to s upport durability and longevity as goals when introducing the prosthetics.
25 Chapter Three Adjustable Compliant Pediatric Knee Adjustability is importance in prosthetic knee design as well as prosthetics in general. No two people are the same shape or size, and have varying needs, abilities and disabilities. Children are no different than adults in this regard and require variability in their prosthetics in order to be properly fitted for their needs. From these criteria, we have utilized a design that offers the simplicity of compliance with the versatility of an adjus table return spring. We have created a way of housing a movable spring that is capable of changing the amount of torque required to bend the knee. By mo ving the position of one end of the adjustable spring along the body of the pr osthetic, variable feedback is possible. From research with Dr. Highsmi th and Dr. Maitland, from the Department of Physical Therapy at the Univ ersity of South Florida, an important concept has made its way into discussion. T he ability to adjust the return spring rate of the lower leg in pediatric prost hetic knees would be an improvement over existing technology. Due to the inherent ly simplistic function of compliant mechanisms a challenge was presented. Dr Highsmith noted t hat a well thought
26 out design would offer the ability to c hange the function and force feedback of the knee from one task to another.  Children, being very energetic and active are often times in situations that require vari able function of their prosthetic. Just like our own knees, which are capable of adapting to our environment instantaneously, children with prosthetic knees need a way of adapting to their environment. For example, a child would re quire different types of feedback from their prosthetic depending on wh ether they were crawling, walking or running. A knee that was suited for crawling would be durable due to constant contact with the ground and flexible with little resistance to bending to allow the child to flex the knee with ease. On t he other hand, a good knee for walking would have to be able to lock during the loading and stance phases of gait. As for running, the prosthetic would have to have a slightly higher resistance to bending to allow the lower leg to keep up with the users stride. For this reason we have developed a way of making the knee adjustable while utilizing the simplicity of compliant design. A static rigid leg is typically prescribed to young children in the early stages of walking to assist them even t hough its only use is for walking. These knees are not able to flex during crawling or sitting and must be removed many times during the day. These knees are often times not used until the child is ready to walk. It was noted by Dr. Highsmith that the longer the child was able to wear his or her prosthetic prior to walk ing the quicker the child would be able to stand. This act of standing is critic al for a childÂ’s mental and physical development. A childÂ’s ability to m anipulate and interact with his or her
27 surroundings increases dramatically from crawling to standing which promotes mental development. A child Â’s environment becomes more interactive once he or she begins to pull up to walk. For this reason it is crucial that prosthetics be made variable and able to be worn for longer periods of time, allowing the child time to adapt prior walki ng.  The physical benefits of early use of prosthetics are achieved through loading of the bones in stance phase. The simple act of standing is a requirement in growing strong bones and joints. The stimulus generated from standing promotes growth within the muscles, bones and the joints. Without these stimuli our legs w ould atrophy and deteriorate over time. As a solution to the problems associated with rigid links, an adjustable compliant four-bar knee was investi gated and improved upon. The basic design utilized was initially conceived by Guri not et al. (2004) . This design was simple and effective but lacked the ability to vary the spring return of the lower limb. For this reason we have developed a new design that allows the user or guardian of the user to change the settings of the knee and alter the response of the output. To do this we took the init ial design and made modifications to the body of the knee to allow for a movable segment within the compliant spring system. This movable spring is what allows the knee to adjust. This is achieved by sliding one end of the spring along the body of the knee to various positions. These various positions allow the spring to be forced into different modes of bending. These different modes of bending have different stiffness associated with them. By increasing t he mode of bending, the force feedback is increased by a linear factor allowing the knee to be adj usted with one simple motion. This
28 added feature creates ability for the knee to lose flexibility when needed and regain it when needed depending on the task at hand.
29 Chapter Four Position/Displacement Analysis This chapter begins by explaining t he significance of the anatomic knee and prosthetic knee function dur ing human gait. The chapt er later talks about how we gathered position data and finishes up with how and why we calculate instant centers. The knee is an important component in walking as well as stance control. The anatomic and polycentric prosthetic knee rotates and translates during swing phase to increase toe clearance and stability during stance. Polycentric knees offer more toe clearance per angle of flex ion than single axis prosthetics. The prosthetic kneeÂ’s shank rotates and trans lates to an angle of 60 with respect to the socket during the end of the initial swing phase and beginning of mid-swing phase to achieve the required amount of toe clearance to prevent stubbing the toe.  In order to determine the ro tational characteristics of the compliant knee, the amount of rotation and transla tion the particular knee designs under go, the shank off each knee was forced to rotate relative to the socket by an amount of, which ranged between 0< <120, in increments of 6. Figure
30 4.1 illustrates the direction of the motion that takes place during the forced rotation of the compliant knee. The rotation of the shank connection is accompanied by a particular am ount of translation, in other words the IC is not fixed but is in constant motion during t he flexion and extension of the prosthetic knee. In Figure 4.1 points A and B indicate the location of the path taken by points on the upper and lower section of th e semi-circle that makes up the lower half of the compliant knee that attaches to the shank. The ICÂ’s location is determined by these paths. Figure 4.1. Rotational plots using finite elem ent analysis, showing the motion of points F and C when the shank is forced to rotate relative to the socket having a bracket angle for = 64
31 In order to calculate t he location of the instant center of the compliant knee, a graphical solution method was utiliz ed. This method is generally used on rigid-body mechanisms because it is relati vely simple to graphically track the motion of the links. The links are constrained to specific paths and it is generally easy to solve for their moti on. By knowing the direction of motion of any two points on a moving object it is possible to fi nd the instant center or rotation of the object. For compliant mechanisms, k nowing where the mechanism will tend to travel at any instant is not easily predi cted. Compliant mechanisms direction of motion is controlled by multiple factors. For single degree of freedom rigid link mechanisms there are only two possible so lutions for any position attained and can be calculated using vector math. Co mpliant mechanisms on the other hand are not as easy to solve. Unlike rigi d link mechanisms which have physical constraints binding their motion, co mpliant mechanisms deform to cope with external forces. Gener ally a pseudo-rigid-body model is utilized to gather information from a compliant mechanism but with the use of finite element analysis we were able to gather the ki nematics data as well as force analysis simultaneously without the need for a pseudo-rigid-body model. First FEA, using two dimensional beam3 elements, was used to accurately plot the position of the knee by placing a pure rotation, i.e. no forces, only torque, on the lower half of the compliant knee at location C in Figure 4.1 The rotation was placed on the knee in steps of six degrees, relative to the socket connection, until a full rotation was reached at 120 degrees. With this dat a we are able to better predict motion
32 characteristics of the knee and offer educated solutions when generating design changes. Figure 4.2. Instantaneous center of rotation path of the compliant knee The data gathered from the rotational displacement was then utilized to find the path of the instant c enter of the shank relative to the socket, shown in Figure 4.2 The perpendicular bisectors A and B of the initial displacement labeled A and B in Figure 4.2 cross one another at the instant center. The perpendicular bisectors are displayed to give a visualization of how we obtain the IC as well as to show where the IC path starts and ends. Four-bar knees on the market also utilize this type of analysis to determine the exact location of the ICÂ’s.
33 The data in that case is used to de termine what changes are needed in the rigid links to obtain an optimum design. An ex ample of IC analysis done on four bar knees is shown in Figure 4.3 This figure shows a lateral view of a typical four bar linkage knee with the instant center plotted at 5-degrees increments of flexion. Points A, B, C and D represent the axes of rotation of the four-bar knee. The IC is determined by intersecting the line through points A and B with the line through points C and D.  Figure 4.3. Instant center path f ound in a four-bar knee design 
34 Of the 11 knees currently on the market for children, all but three of the knees are four-bar knees. Four-bar knees provide greater toe clearance than single-axis knees for a given knee-flexion angle and offer stance-phase stability. This added toe clearance allows the user to walk with less concern for stubbing their toes and tripping during the gait cycl e. During the gait cycle, as much as 3.2 additional centimeters of toe clearance is achieved ov er single axis knees in adult testing.  In addition to the added toe clearance ac hieved, the four-bar knee also offers stance-phase stability. The stabilit y of a four-bar knee during load-bearing is determined by the location of the IC with respect to the ground reaction force vector. Prosthetists are given some cont rol over the degree of stability through prosthetic alignment. For stability during stance, the IC must be posterior to the ground reaction force to maintain the ext ension moment which keeps the knee in a locked upright position, illustrated in Figure 1.1 with the exception of the toe off phase.  From the pos ition analysis done on the compliant knee we have found that the IC falls well behind what would be the ground reaction force in simulation. This data proves the compli ant knee is stable based on the instant center location.
35 Conclusion By knowing the ICÂ’s path for the full rotati on of the compliant knee, it is possible to compare this information with the dat a found on IC paths of current four-bar knees. If a light weight compliant knee can match or improve on the kinematic behavior of a rigid link knee, it will be a benefit to pediatric amputees. Future areas for optimization of co mpliant knees incl ude examining the effect on the instant center caused by varying specific parameters. These parameters include the compliant spring l ength, bracket angles and compliant spring material. Simulations could possibly assist in answering these questions. In addition to not knowing how the knee will react to dynamic loads, we feel we need to research where the IC would best be located to offer the most assistance to the user. Although there is significant data available on the proper IC path for adult four-bar knees, less is known about w hat a child would benefit from. We wish to know if it is possible to relate the motion of the compliant knee back to rigid-body four-bar motion. Future work also includes creati ng a simulation of the compliant knee function throughout the gait cycle, in other words there is a particular combination of forces and moments that will be seen by the knee during the amputeeÂ’s gait. Thus far we have only simulated linear and rotational displacements, not a realistic combination of force and moments.
36 The first step in obtaining realistic fo rce and moment combinations is to use ground reaction data from a rigid link kn ee. It is not yet certain that such data would faithfully represent the forces seen by the compliant knee but this has to be the starting point. Previous research on rigid link knees has indicated that it is desirable for the instant center to be lo cated near a sound joint or th e end of the residual limb. We hypothesize that this will still be tr ue and it appears to be the case in the designs researched thus far. Another research question is whet her the same design philosophy as used for adult knees truly applies in the pedi atric knee. Certainly the laws of physics apply equally well to adults and child ren. However, the pediatric knee serves as a training function rather than a re-training function. The child learning to walk does not have prior experienc e. Thus, the knee may need to be designed to actively discourage poor walkin g habits by making them either more comfortable or less tiring to use. It may also be desirable to index the motion of the instant centers in a precise way. One possible approach to this problem would be to determine the rigid four-bar link with instant center mo tion that most resembles the compliant four-bar motion.
37 Chapter Five Force and Stress Analysis In this chapter, the force and stress anal ysis of the adjustabl e compliant knee is discussed. First, investigations into the stiffness of the knee with respect to applied loads are described. Then, the st ress is analyzed to validate the kneeÂ’s ability to function under prescribed loads Lastly, calculations were made to acquire a safety factor for the compli ant knee based on peak forces associated with normal gait. To better understand how the knee would r eact to the weight of a child, we studied the force characteristics of the knee throughout a full range of motion for multiple bracket configurations. T he adjustable brackets, illustrated in Figure 5.1, offer a way to adjust the compliant segm ents by moving the location of brackets. The bracket angle illustrated in Figure 5.1 ranges from 0< <140. To further analyze the kneeÂ’s force feedback we took the data from the displacement analysis to give us a baseline starting pos ition for our new displacement paths. This baseline starting positi on is the path of lowest resistance to rotation for the mechanism. In order to build on the work done in the previous chapter, the data
38 about the path of lowest resistance is used to determine the resistance of the mechanism to motion away from that path. This is done by applying displacements first along the path of leas t resistance and then in a zigzag pattern about that path as shown in Figure 5.2. For each individual displacement along the path made, the knee was forced to move away from the path of least resistance in a way that would cover as much area as possible. To prevent missing data locations, the force magnit ude data was overlapped as illustrated in Figure 5. 2 using a log10 scale. The data from the fo rce plots was later smoothed using interpolation between the adjac ent data points, as is shown in part a) of Figures 5.3-5.6 The length of the vector arrows in part a) of each of Figures 5.35.6 indicate the magnitude of the force being applied to the knee at each data location. Because the magnit udes of the forces are very large in the lower left hand side of the plot, the forc es at other locations are proportionally too small to see. In part b), of Figures 5.3-5.6, the arrows are normalized to show the force directions and do not represent the act ual magnitude. These steps were then repeated for multiple bracket angle conf igurations to gather data on the change in force feedback due to the angle devia tion in the compliant spring.
39 Figure 5.1. Prototype design labeled -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 -0.04-0.03-0.02-0.0100.010.02 Force Plot Path Path of Lowest Energy Figure 5.2. Paths used to generate force plots measured in meters
40 Figure5.3. Force plot data of the compliant knee in kg on a log10 scale utilizing overlapping of multiple data sets with a bracket angle of 50 degrees a) b) Figure 5.4. Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 80 degrees measured in meters
41 a) b) Figure 5.5. Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 70 degrees measured in meters a) b) Figure 5.6. Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 60 degrees measured in meters a) b) Figure 5.7. Force plot data of t he compliant knee in kg on a log10 scale with a bracket angle of 50 degrees measured in meters
42 From the force plots we generate useful information about how the compliant knee will respond to external forces. The plots illustrated in Figures 5.3-5.6 clearly show the lowest forces fall along the path of least resistance and is illustrated in the figures. The higher forces in Figures 5.3-5.6 are plotted using red while lower forces appear as blue. From these plots we know the forces increase rapidly away from the path of lowest energy indicating a relatively high resistance to deflection away from the in tended path. In other words, the knee requires large forces to displace the shank connection, shown in Figure 4.1, from its path of least resistance. This is important because excessive displacements away from the projected pat h could result in instabi lity and could potentially cause the amputee to fall. From inspecti on, forces increase more rapidly when the knee is in compression. This is largel y due to the tension that results in the shorter compliant segment, labeled D in Figure 4.1. This result is the case because the shorter compliant segment is mostly in tensile axial happens to be stiffer than the bending mode seen along th e path of lowest resistance. Simultaneous to the FEA force data being calculated, the stress analysis data was also calculated. The plots shown in Figure 5.8-5.11 illustrate the ultimate stress endured by the compliant kn ee as it was forced to plot along the path shown in Figure 5.2.
43 Figure 5.8. Stress plot data of t he compliant knee in GPa on a log10 scale with a bracket angle of 80 degrees measured in meters Figure 5.9. Stress plot data of t he compliant knee in GPa on a log10 scale with a bracket angle of 70 degrees measured in meters
44 Figure 5.10. Stress plot data of the compliant knee in GPa on a log10 scale with a bracket angle of 60 degrees measured in meters Figure 5.11. Stress plot data of the compliant knee in GPa on a log10 scale with a bracket angle of 50 degrees measured in meters
45 Conclusion The stress plots look surprisingly similar to the force plots with the exception of the magnitude of the numbers present in the key. The values of stress are significantly higher than the force. The force and stress plots indicate t hat the knee woul d reach equilibrium at .02 inches if loaded with a point load of 200 lbs in stance. Unfortunately the stress in the compliant members for a 200 pound load reached approximately 3.5 GPa, nearly 2 times the acceptable t ensile strength of 1.758 GPa for SAE 10701090 high carbon tempered spring steel, i ndicating a failure would result.  The color plot color that corresponds to 1.758 GPa, which translates to 9.25 on the log10 scale, is displayed as light blue shown in Figure 5.11. The exceeded stress levels indicate that modification will need to be made to the design in order to overcome the stresses acquired during the test load. For these tests it is assumed that the ends of compliant members have fixed position and angle using beam3 type elements. Figure 5.12. Corresponding plot color for 1.758 GPa on a log10 scale
46 Future work involves plotting precis e regions of acceptable stress levels for the knee. This would allow us to better determine how much deflection would be required to reach the stress limitations for various positions and could be used to calculate the exact force required to generate these stress levels. Future areas for optimization of co mpliant knees incl ude examining the change in forces and stresses within the knee when changing the following parameters. Spring lengths Angles of the spring Stiffness of the springs Contact elements
47 Chapter Six Discussion and Conclusions We have designed and examined a pediat ric compliant knee mechanism that may offer solutions to problems that exis t for young children who are just learning to walk. One of our achievements was in mechanism design by way of creating a pediatric compliant knee mechanism with the ability to adjust ex ternal rotational resistance. In addition to the func tional achievements, we have created a prototype weighing le ss than five ounces. We formu lated a method for analyzing the rotation and translational motion using nonlinear FEA of the compliant knee accomplished through calculation and plotting of the instant centers of the lower shank with respect to the socket connecti on. In addition, we have developed a method for simultaneously calc ulating the external reaction forces and internal stresses present for displacements made wit hin the anticipated region of motion. From our analysis position, force and stre ss data was gathered for a deflection of for 0< <120. The instant centers and protot ype both indicate that the knee offers adequate stability for stance loading. Unfortunately the force and stress plots indicate that the knee will not suppor t a point load of 200 lbs in. The stress in the compliant members for a 200 pound load reached approxim ately 3.5 GPa,
48 twice the acceptable tens ile strength of 1.758 GP a for SAE 1070-1090 high carbon tempered spring steel.  The exceeded stress levels indicate that modification will need to be made to the desi gn in order to overcome the stresses acquired during the test load. Future work includes: Design a gait simulator for FEA Research the effect of IC location Find the IC location that would benefit children index the motion of the inst ant centers in a precise way Find and relate a four-bar knee that most resembles the compliant fourbar motion Introduce contact elements that will redirect forces through the rigid frame of the knee and away from the compliant members
49 References 1. Highsmith, Michael, OPT. Personal communication. 2. Wheeless' Textbook of Orthopaedics (2000). Duke University. Retrieved October 11, 2007 http://www.wheelessonline.co m/ortho/stance_phase_of_gait 3. Pitkin, M. (1997). Effects of Design Variants in Lower-Limb Prostheses on Gait Synergy. Journal of Prosthetics and Orthotics, 9( 3), pp. 113-122. 4. Dundass, C., Zao, G. and Mechefske, C. (2003). Initial Biomechanical Analysis and Modeling of Transfemoral Amputee Gait. Journal of Prosthetics and Orthotics, 15(1), pp. 20-26. 5. Muller, Werner (1996, November-Decembe r). Form and function of the knee: its relation to high performance and to spor ts. The American Journal of Sports Medicine, 24(6), S104(3). 6. Johnson, M., Schmidt, R., Solomon, E. and Davis, P. (1985). Human Anatomy. New York, NY: Saunders College Publishing. 7. Webster's Revised Unabr idged Dictionary (1913). 8. Wilk, b., Karol, L., Halliday, S., Cu mmings, D., Haideri, N. and Stephenson, J. (1999). Transition to an Articulating Knee Prosthesis in Pediatric Amputees, 11(3), pp. 69-74. 9. Andrysek, Jan, Naumann, Stephen, Cleghorn William. Â“Design Characteristics of Pediatric Prosthetic KneesÂ” December 2004 http://ieeexplore.ieee.org/iel5/73 33/29923/01366424.pdf#s earch=%22pediatric% 20prosthetic%20knee%20technology%22. 10. Howell, Brigham Young University Compliant Mechanisms Research, (2004) http://research.et.byu.edu/ llhwww/intro/intro.html
50 11. Gurinot, E., Alexandre, Magleby, P., S pencer, Howell, L., La rry, Preliminary Design Concepts for Compliant Mechanism Prosthetic Knee Joints, October 2, 2004. 12. Ayyappa, Edmond Â“Normal Human Locomot ion, Part 2: Motion, Ground Reaction Force and Muscle ActivityÂ” JPO 1997; Vol 9, Num 2, pp. 42-57. 13. Gard, Steven, Childress, Dudley, Uelle ndahl, Jack Â”The Influence of Four-Bar Linkage Knees on Prosthetic Swing-Phase Floor Clear anceÂ” JPO 1996; Vol 8, Num 2, pp. 34 14. Smalley Steel Ring Company, (2007) http://pdf.directindustry.com/pdf/smalle y/engineering-parts-catalog-metric/118127507-_87.html
52 Appendix A Ansys Code Used to Find Force and Stress Data Modify the value Â“theta4degreesÂ” to attain the force and stress data knee.bat ! This ansys batch file analyzes compliant four-bar knee designs /COM, /COM,Preferences for GUI filt ering have been set to display: /COM, Structural /COM, Thermal !* !* !******************BEGIN PREPROCESSOR STEPS *********** /PREP7 !******************************************************* !************SET UP MODEL PARAMETERS******************** !******************************************************* !********** UNITS IN NEWTONS, METERS, ETC.
53 Appendix A (Continued) !*** !*** !--------------number of divisions-------------------Segments3 =120 Segments5 = Segments3/3 !--------------------Variables-----------------------theta4degrees = 45 !---------------Calculate parameters-----------------in_m = .0254 PI = acos(-1.) R = in_m*1.02 Knee arc radius ---> 1.02 inches theta2 = 176.5*PI/180 first comp liant beam origin on knee arc in radians L3 = in_m*.5 first compliant segment length ---> .5 inch theta3 = 300*PI/180 b3ang = 275*PI/180 bracket angle for first compliant segment theta4 = theta4degrees*PI/180 second compli ant segment origin on knee arc in radians L5 = in_m*(1.5) second compliant segement length ----> 1.5 inch theta5 = theta4+165*PI/180 b5ang = 265*PI/180 bracket angle for second compliant segment Lleg = 4*in_m guess lower leg mass center at 4 inches from center of knee
54 Appendix A (Continued) delta3 = .625*in_m assembly distance for point 5 from point 3 x3 = R*cos(theta2)+L3*cos(theta3) y3 = R*sin(theta2)+L3*sin(theta3) x5 = R*cos(theta4)+L5*cos(theta5) y5 = R*sin(theta4)+L5*sin(theta5) dx5 = x3 +delta3 x5 displacement in x for point 5 dy5 = y3 y5 displacement in y for point 5 t2 = theta2*180/PI t3 = theta3*180/PI t4 = theta4*180/PI t5 = theta5*180/PI b3 = b3ang*180/PI b5 = b5ang*180/PI d3 = delta3/in_m !---------------file name subscript---------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! define segment properties !!!! compliant segments !!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
55 Appendix A (Continued) t1 = 1/200*in_m thickness is 1/200 inch w1 = 1.5*in_m width is 1.5 inch A1 = t1*w1 Area I1 = w1*t1*t1*t1/12 Moment of Inertia Esteel = 200E9 Young's Modulus in GPa psteel = 0.28 poisson's ratio !* rigid segments t2 = 0.5*in_m thickness is 1/200 inch w2 = 1.5*in_m width is 1.5 inch A2 = t2*w2 Area I2 = w2*t2*t2*t2/12 Moment of Inertia Ealuminum = 70E9 Young's Modulus in GPa paluminum = 0.38 poisson's ratio ET,1,BEAM3 KEYOPT,1,6,1 KEYOPT,1,9,0 KEYOPT,1,10,0 R,1,A1,I1,t1, , R,2,A2,I2,t2,0,0,0, MPTEMP,,,,,,,, MPTEMP,1,0 MPDE,EX,1
56 Appendix A (Continued) MPDE,PRXY,1 MPDATA,EX,1,,Esteel MPDATA,PRXY,1,,psteel MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,2,,Ealuminum MPDATA,PRXY,2,,paluminum !-----------CREATE KEYPOINTS: K,Point #, X-coord, Y-COORD, Z-Coord K,1,0,0,0 K,2,R*cos(theta2),R*sin(theta2),0 K,3,R*cos(theta2)+L3*cos(theta3), R*sin(theta2)+L3*sin(theta3),0 K,4,R*cos(theta4),R*sin(theta4),0 K,5,R*cos(theta4)+L5*cos(theta5), R*sin(theta4)+L5*sin(theta5),0 K,6,R,0,0 K,7,-R,0,0 K,8,R+Lleg,0,0 !----------Create knee usi ng arcs and lines-------! arcs in the knee: 7 to 2, 2 to 4, and 4 to 6 LARC,7,2,1,R LESIZE,1,,,5 LARC,2,4,1,R LESIZE,2,,,5
57 Appendix A (Continued) LARC,4,6,1,R LESIZE,3,,,5 lines in model: 2 to 3, 4 to 5, and 6 to 8 L,2,3 LESIZE,4,,,Segments3 L,4,5 LESIZE,5,,,Segments5 L,6,8 LESIZE,6,,,10 !-------------------MESH -----------type,1 real,1 mat,1 LMESH,4 LMESH,5 real,2 mat,2 LMESH,1 LMESH,2 LMESH,3 LMESH,6 FINISH
58 Appendix A (Continued) !-----------Get Nodes at Chosen Keypoints----------ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max nsel,all ksel,all ksel,s,kp,,5 nslk,s *get,nkp5,node,0,num,max nsel,all ksel,all ksel,s,kp,,6 nslk,s *get,nkp6,node,0,num,max nsel,all ksel,all ksel,s,kp,,7 nslk,s *get,nkp7,node,0,num,max nsel,all
59 Appendix A (Continued) ksel,all !******************************************* !***********SOLUTION STEPS ***************** !******************************************* /SOLU Set to Nonlinear Deflection Analysis NLGEOM,on CNVTOL,ROT,,0.01,,0 CNVTOL,U,,0.01,,0 CNVTOL,F,,0.001,,0 Set Analysis Type to Static (0) ANTYPE, 0 !**************************************************************** !******************* Set up Boundary Conditions ***************** !**************************************************************** !-----------------------------------------------| Keypoint3 is fixed with a twist on the end | !-----------------------------------------------|
60 Appendix A (Continued) DK,3,UX,0,, DK,3,UY,0,, DK,3,ROTZ,b3ang-theta3,, DK,5,UX,dx5,, DK,5,UY,dy5,, DK,5,ROTZ,b5ang-theta5,, steps = 0 steps = steps+1 lswrite,steps *enddo lssolve,steps move1=steps !************************************************************************ ________________________ Displacement Input _________________________ !************************************************************************ *do,nn,1,5,1 DK,6,UX,.005*nn Dk,6,UY,0 steps=steps+1 lswrite,steps disx11 = .005*nn
61 Appendix A (Continued) *enddo lssolve,move1+1,steps move2=steps *DO,ll,1,10,1 DK,6,UX,disx11 Dk,6,UY,-.005*ll steps=steps+1 lswrite,steps dis22y = -.005*ll *ENDDO lssolve,move2+1,steps move3=steps *DO,kk,0,10,2 *DO,rr,1,10,1 DK,6,UX,disx11-rr*.005 Dk,6,UY,dis22y+kk*.005 steps=steps+1 lswrite,steps disx33 = disx11-rr*.005 *ENDDO lssolve,move3+1,steps move4 = steps
62 Appendix A (Continued) *DO,ss,0,9,1 DK,6,UX,disx33+ss*.005 Dk,6,UY,dis22y+kk+1*005 steps=steps+1 lswrite,steps *ENDDO lssolve,move4+1,steps move5 = steps steps = steps+1 lswrite,steps *ENDDO lssolve,move5+1,steps FINISH
63 Appendix A (Continued) !**************************************************************** ++++++++++++++++++++ Results ++++++++++++++++++++++++ !**************************************************************** /POST1 *DIM,Smax,TABLE,Steps *DIM,Ydis7,TABLE,Steps *DIM,Xdis7,TABLE,Steps *DIM,Ydis6,TABLE,Steps *DIM,Xdis6,TABLE,Steps *DIM,Qdis6,TABLE,Steps !INPUT ROTATION ON NODE 6 *DIM,Xforce6,TABLE,Steps *DIM,Yforce6,TABLE,Steps *DIM,Xforce3,TABLE,Steps *DIM,Yforce3,TABLE,Steps *DIM,Xforce5,TABLE,Steps *DIM,ZMoment5,TABLE,Steps *DIM,ZMoment3,TABLE,Steps *DIM,Yforce5,TABLE,Steps *DO,mm,1,steps,1
64 Appendix A (Continued) SET,mm ETABLE,smxi,NMIS,1 ESORT,ETAB,smxi,0,1 *GET,stress,SORT,0,MAX *SET,Smax(mm),stress *GET,disX7,NODE,nkp7,U,X *SET,Xdis7(mm),disX7 *GET,disY7,NODE,nkp7,U,Y *SET,Ydis7(mm),disY7 *GET,disY6,NODE,nkp6,U,Y *SET,Ydis6(mm),disY6 *GET,disX6,NODE,nkp6,U,X *SET,Xdis6(mm),disX6 *GET,disQ6,NODE,nkp6,ROT,Z *SET,Qdis6(mm),disQ6 *GET,forceX3,Node,nkp3,RF,FX *SET,Xforce3(mm),forceX3 *GET,forceY3,Node,nkp3,RF,FY *SET,Yforce3(mm),forceY3 !*GET,forceX6,Node,nkp6,RF,FX !*SET,Xforce6(mm),forceX6 !*GET,forceY6,Node,nkp6,RF,FY !not needed, Fxy = 0 when rotating !*SET,Yforce6(mm),forceY6 *GET,MomentZ3,Node,nkp3,RF,MZ
65 Appendix A (Continued) *SET,ZMoment3(mm),MomentZ3 *GET,forceX5,Node,nkp5,RF,FX *SET,Xforce5(mm),forceX5 *GET,forceY5,Node,nkp5,RF,FY *SET,Yforce5(mm),forceY5 *GET,MomentZ5,Node,nkp5,RF,MZ *SET,ZMoment5(mm),MomentZ5 *ENDDO /output,knee_rotation_% theta4degrees%deg,txt,,Append *MSG,INFO1,'Theta2','Theta3','BrAng3 ','Theta4','Theta5','BrAng5','Delta3' %-9C %-9C %-9C %-9C %-9C %-9C %-9C *VWRITE,t4,t2,t3,b3,t5,b5,d3 %9.2G %9.2G %9.2G %9.2G %9.2G %9.2G %9.2G *MSG,INFO2,'Stress_Max','Ydis7', 'Xdis7','Xdis6', 'Ydis6','Rot6' %-11C %-11C %-11C %-11C %-11C %-11C *VWRITE,Smax(1),Ydis7(1),Xdis7(1),Xdis6(1),Ydis6(1),Qdis6(1) %11.4G %11.4G %11.4G %11.4G %11.4G %11.4G *MSG,INFO2,'Xforce3','Yforce3','Xfo rce5','Yforce5','ZMoment3','ZMoment5' %-11C %-11C %-11C %-11C %-11C %-11C *VWRITE,Xforce3(1),Yforc e3(1),Xforce5(1),Yforce5(1), ZMoment3(1), ZMoment5(1) %11.4G %11.4G %11.4G %11.4G %11.4G %11.4G !*MSG,INFO2,'Xforce6','Yforce6'
66 Appendix A (Continued) %-11C %-11C !*VWRITE,Xforce6(1),Yforce6(1)! not needed, fxy = 0 when rotating %11.4G %11.4G /output FINISH !**************************************************** ************ PLOT FINAL POSITION ************ !**************************************************** /POST1 /EFACE,1 SET,LoadSteps AVPRIN,0,0, PLNSOL,U,X,1,1 FINISH /POST1 PLDISP,1
67 Appendix A (Continued) Ansys Output File Theta2 Theta3 BrAng3 Theta4 Theta5 BrAng5 Delta3 64. 1.27E-02 3.00E+02 2.75E+02 2.29E+02 2.65E+02 0.63 Appendix A (Continued) Stress_Max Ydis7 Xdis7 Xdis6 Ydis6 Rot6 4.5919E+08 1.0575E-03 2.5879E03 2.5641E-03 -5.1223E-04 -3.0298E-02 4.5919E+08 1.0575E-03 2.5879E03 2.5641E-03 -5.1223E-04 -3.0298E-02 5.6029E+08 9.6236E-04 2.3375E03 2.3297E-03 5.8006E-05 -1.7453E-02 4.9449E+08 1.1950E-03 2.9858E03 2.9147E-03 -1.5168E-03 -5.2360E-02 6.6028E+08 1.3574E-03 3.5262E03 3.3290E-03 -3.1585E-03 -8.7266E-02 7.7049E+08 1.4718E-03 3.9684E -03 3.5821E-03 -4.8428E-03 -0.1222 8.3983E+08 1.5559E-03 4.3377E -03 3.6998E-03 -6.5497E-03 -0.1571 8.8151E+08 1.6208E-03 4.6579E -03 3.7059E-03 -8.2659E-03 -0.1920 9.0469E+08 1.6726E-03 4.9454E -03 3.6173E-03 -9.9832E-03 -0.2269 9.1519E+08 1.7147E-03 5.2106E -03 3.4450E-03 -1.1696E-02 -0.2618 9.2888E+08 1.7490E-03 5.4600E -03 3.1959E-03 -1.3400E-02 -0.2967 9.9700E+08 1.7768E-03 5.6977E -03 2.8747E-03 -1.5093E-02 -0.3316 1.0607E+09 1.7988E-03 5.9263E -03 2.4847E-03 -1.6770E-02 -0.3665 1.1203E+09 1.8156E-03 6.1473E -03 2.0282E-03 -1.8430E-02 -0.4014 1.1761E+09 1.8277E-03 6.3620E -03 1.5073E-03 -2.0070E-02 -0.4363
68 Appendix A (Continued) 1.2281E+09 1.8354E-03 6.5708E -03 9.2328E-04 -2.1688E-02 -0.4712 1.2767E+09 1.8390E-03 6.7743E -03 2.7763E-04 -2.3282E-02 -0.5061 1.3218E+09 1.8388E-03 6.9725E -03 -4.2843E-04 -2.4848E-02 -0.5411 1.3653E+09 1.8351E-03 7.1656E -03 -1.1938E-03 -2.6386E-02 -0.5760 1.4076E+09 1.8281E-03 7.3535E -03 -2.0172E-03 -2.7892E-02 -0.6109 1.4470E+09 1.8183E-03 7.5362E -03 -2.8976E-03 -2.9365E-02 -0.6458 1.4835E+09 1.8059E-03 7.7135E -03 -3.8338E-03 -3.0803E-02 -0.6807 1.5172E+09 1.7911E-03 7.8852E -03 -4.8247E-03 -3.2203E-02 -0.7156 1.5533E+09 1.7744E-03 8.0512E -03 -5.8689E-03 -3.3564E-02 -0.7505 1.5868E+09 1.7560E-03 8.2112E -03 -6.9652E-03 -3.4883E-02 -0.7854 1.6176E+09 1.7363E-03 8.3651E -03 -8.1123E-03 -3.6159E-02 -0.8203 1.6504E+09 1.7156E-03 8.5128E -03 -9.3087E-03 -3.7390E-02 -0.8552 1.6830E+09 1.6943E-03 8.6540E -03 -1.0553E-02 -3.8574E-02 -0.8901 1.7133E+09 1.6726E-03 8.7886E -03 -1.1844E-02 -3.9709E-02 -0.9250 1.7494E+09 1.6509E-03 8.9164E -03 -1.3179E-02 -4.0794E-02 -0.9599 1.7832E+09 1.6295E-03 9.0375E -03 -1.4557E-02 -4.1827E-02 -0.9948 1.8233E+09 1.6087E-03 9.1516E03 -1.5977E-02 -4.2806E-02 -1.030 1.8630E+09 1.5889E-03 9.2589E03 -1.7436E-02 -4.3730E-02 -1.065 1.9125E+09 1.5701E-03 9.3592E03 -1.8933E-02 -4.4598E-02 -1.100 1.9657E+09 1.5528E-03 9.4526E03 -2.0465E-02 -4.5408E-02 -1.134 2.0258E+09 1.5370E-03 9.5391E03 -2.2030E-02 -4.6159E-02 -1.169 2.1000E+09 1.5231E-03 9.6190E03 -2.3627E-02 -4.6851E-02 -1.204 2.1854E+09 1.5111E-03 9.6924E03 -2.5254E-02 -4.7481E-02 -1.239 2.2845E+09 1.5012E-03 9.7594E03 -2.6907E-02 -4.8050E-02 -1.274
69 Appendix A (Continued) 2.3990E+09 1.4935E-03 9.8204E03 -2.8584E-02 -4.8556E-02 -1.309 2.5349E+09 1.4880E-03 9.8756E03 -3.0284E-02 -4.8999E-02 -1.344 2.6904E+09 1.4848E-03 9.9254E03 -3.2003E-02 -4.9379E-02 -1.379 2.8607E+09 1.4839E-03 9.9703E03 -3.3739E-02 -4.9694E-02 -1.414 3.0589E+09 1.4854E-03 1.0011E02 -3.5490E-02 -4.9944E-02 -1.449 3.2690E+09 1.4891E-03 1.0047E02 -3.7252E-02 -5.0129E-02 -1.484 3.5053E+09 1.4952E-03 1.0081E02 -3.9023E-02 -5.0249E-02 -1.518 3.7633E+09 1.5034E-03 1.0111E02 -4.0799E-02 -5.0304E-02 -1.553 4.0401E+09 1.5140E-03 1.0140E02 -4.2579E-02 -5.0293E-02 -1.588 4.3384E+09 1.5267E-03 1.0168E02 -4.4359E-02 -5.0218E-02 -1.623 4.6608E+09 1.5417E-03 1.0195E02 -4.6136E-02 -5.0076E-02 -1.658 5.0109E+09 1.5589E-03 1.0223E02 -4.7906E-02 -4.9870E-02 -1.693 5.3925E+09 1.5785E-03 1.0253E02 -4.9667E-02 -4.9599E-02 -1.728 5.8105E+09 1.6005E-03 1.0285E02 -5.1415E-02 -4.9263E-02 -1.763 6.2704E+09 1.6250E-03 1.0322E02 -5.3148E-02 -4.8862E-02 -1.798 6.7788E+09 1.6521E-03 1.0364E02 -5.4860E-02 -4.8397E-02 -1.833 7.3437E+09 1.6819E-03 1.0413E02 -5.6549E-02 -4.7869E-02 -1.868 7.9742E+09 1.7143E-03 1.0470E02 -5.8212E-02 -4.7278E-02 -1.902 8.6810E+09 1.7491E-03 1.0537E02 -5.9843E-02 -4.6624E-02 -1.937 9.4764E+09 1.7859E-03 1.0616E02 -6.1441E-02 -4.5910E-02 -1.972 1.0374E+10 1.8242E-03 1.0708E02 -6.3000E-02 -4.5136E-02 -2.007 1.1390E+10 1.8631E-03 1.0814E02 -6.4518E-02 -4.4305E-02 -2.042 Xforce3 Yforce3 Xforce5 Yforce5 ZMoment3 Zmome
70 Appendix A (Continued) 0.9787 -0.3347 -0.9787 0.3347 -4.7121E-02 4.1809E-02 0.9787 -0.3347 -0.9787 0.3347 -4.7121E-02 4.1809E-02 1.972 1.705 -1.972 -1.705 -5.7641E-02 4.4153E-02 -0.7198 -3.427 0.7198 3.427 -3.0569E-02 3.6918E-02 -3.211 -7.200 3.210 7.198 -8.5401E-03 2.7942E-02 -5.325 -9.748 5.325 9.747 8.8893E-03 1.8915E-02 -7.074 -11.38 7.074 11.38 2.2877E-02 1.0837E-02 -8.533 -12.39 8.533 12.39 3.4502E-02 4.0528E-03 -9.774 -12.98 9.774 12.98 4.4524E-02 -1.4188E-03 -10.85 -13.27 10.85 13.27 5.3428E-02 -5.6685E-03 -11.80 -13.35 11.80 13.35 6.1516E-02 -8.8052E-03 -12.64 -13.28 12.64 13.28 6.8982E-02 -1.0928E-02 -13.40 -13.08 13.40 13.08 7.5948E-02 -1.2118E-02 -14.07 -12.79 14.07 12.79 8.2492E-02 -1.2443E-02 -14.67 -12.42 14.67 12.42 8.8667E-02 -1.1953E-02 -15.20 -12.00 15.20 12.00 9.4505E-02 -1.0690E-02 -15.67 -11.53 15.67 11.53 0.1000 -8.6849E-03 -16.07 -11.02 16.07 11.02 0.1052 -5.9644E-03 -16.40 -10.49 16.40 10.49 0.1102 -2.5482E-03 -16.67 -9.941 16.67 9.941 0.1148 1.5478E-03 -16.88 -9.386 16.88 9.386 0.1191 6.3108E-03 -17.02 -8.831 17.02 8.832 0.1231 1.1731E-02 -17.10 -8.286 17.10 8.287 0.1268 1.7800E-02 -17.12 -7.760 17.12 7.760 0.1302 2.4512E-02
71 Appendix A (Continued) -17.07 -7.260 17.07 7.260 0.1333 3.1864E-02 -16.97 -6.796 16.97 6.796 0.1360 3.9852E-02 -16.81 -6.377 16.81 6.377 0.1384 4.8475E-02 -16.59 -6.012 16.59 6.012 0.1405 5.7734E-02 -16.32 -5.710 16.32 5.711 0.1422 6.7632E-02 -16.00 -5.483 16.01 5.484 0.1436 7.8173E-02 -15.64 -5.340 15.64 5.340 0.1447 8.9366E-02 -15.24 -5.292 15.25 5.292 0.1455 0.1012 -14.81 -5.350 14.81 5.351 0.1459 0.1137 -14.35 -5.527 14.35 5.528 0.1461 0.1270 -13.87 -5.836 13.88 5.837 0.1459 0.1409 -13.39 -6.291 13.39 6.292 0.1455 0.1556 -12.89 -6.907 12.89 6.908 0.1448 0.1710 -12.41 -7.700 12.41 7.702 0.1439 0.1873 -11.95 -8.691 11.95 8.692 0.1429 0.2044 -11.52 -9.900 11.52 9.901 0.1417 0.2225 -11.14 -11.35 11.14 11.35 0.1404 0.2415 -10.82 -13.07 10.82 13.07 0.1390 0.2616 -10.59 -15.10 10.59 15.10 0.1377 0.2829 -10.46 -17.47 10.46 17.47 0.1365 0.3055 -10.47 -20.22 10.47 20.22 0.1354 0.3295 -10.64 -23.42 10.64 23.42 0.1347 0.3550 -11.01 -27.13 11.01 27.13 0.1343 0.3823 -11.63 -31.43 11.63 31.43 0.1344 0.4116
72 Appendix A (Continued) -13.84 -42.23 13.84 42.23 0.1367 0.4772 -15.58 -49.01 15.58 49.01 0.1393 0.5143 -17.90 -56.97 17.90 56.97 0.1431 0.5547 -20.91 -66.36 20.91 66.36 0.1485 0.5991 -24.83 -77.51 24.83 77.51 0.1556 0.6481 -29.88 -90.85 29.88 90.85 0.1651 0.7026 -36.40 -107.0 36.40 107.0 0.1772 0.7636 -44.84 -126.6 44.84 126.6 0.1926 0.8325 -55.82 -150.8 55.82 150.8 0.2119 0.9108 -70.23 -181.1 70.23 181.1 0.2361 1.001 -89.34 -219.4 89.33 219.4 0.2662 1.105 -115.0 -268.8 115.0 268.8 0.3036 1.228 Matlab Code for Plotting Force and Stress Plots Modify the Â“textÂ” value to chang e files used to generate the plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% % Ansys data analysis file % % For an Ansys batch file % % which produces an output file named knee_output.txt % % Version 1: May 18,2007 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%
73 Appendix A (Continued) clear all text = '50deg' type = 'interp' % type = 'flat' filename = ['knee_force_plot_',text,'.txt']; string1 = ['C:\DOCUME~1\smahler\MYDOC U~1\lusk_ansys\ansys_output_files\'] fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT fclose(fid1); %closes the data file GBT = native2unicode(ABT)'; %changes data from machine code to text s_iB = findstr('Rot6', GBT); % finds end of header s_if = findstr('Xforce6', GBT); % finds end of header s_iB2 = findstr('Yforce6', GBT); % finds end of header A=str2num(GBT(s_iB(end)+5:s_if(end)-1))% turns the data into a numerical matrix A2=str2num(GBT(s_iB2(end)+9:end))% turn s the data into a numerical matrix %figure(1) %p lot(90-A(1:50,1)*180/pi,A(1:50,2)) %plot(A(:,1)) %axis equal %title('stress') figure(1) clf
74 Appendix A (Continued) x=A(:,4)' y=A(:,5)' % u_sign = sign(A2(:,1)'); % v_sign = sign(A2(:,2)'); % u=log10(abs(A2(:,1)')).*u_sign; % v=log10(abs(A2(:,2)')).*u_sign; u=A2(:,1)'; v=A2(:,2)'; %C=(A(:,2).^2+A(:,3).^2).^.5; xp = ; yp= ; vp=; up=; xp1 = ; yp1= ; vp1=; up1=; xp2 = ; yp2= ; vp2=; up2=;
75 Appendix A (Continued) xp3 = ; yp3= ; vp3=; up3=; xp4 = ; yp4= ; vp4=; up4=; %1st run for i = 1:5, ls = 22 sv = 2 xp4 = [xp4;[x(ls*(i-1)+5+sv:ls*i+sv-6-1)]] yp4 = [yp4;[y(ls*(i-1)+5+sv:ls*i+sv-6-1)]] vp4 = [vp4;[v(ls*(i-1)+5+sv:ls*i+sv-6-1)]] up4 = [up4;[u(ls*(i-1)+5+sv:ls*i+sv-6-1)]] end %2nd run for i = 1:6, ls = 32 sv =115 xp = [xp;[x(ls*(i-1)+sv+24:-1:ls*(i -1)+sv+9)];[x(ls*(i -1)+sv+25:ls*(i1)+sv+32)],[x(ls*(i)+sv+1:ls*(i)+sv+8)]]
76 Appendix A (Continued) yp = [yp;[y(ls*(i-1)+sv+24:-1:ls*(i -1)+sv+9)];[y(ls*(i -1)+sv+25:ls*(i1)+sv+32)],[y(ls*(i)+sv+1:ls*(i)+sv+8)]] up = [up;[u(ls*(i-1)+sv+24:-1:ls*(i -1)+sv+9)];[u(ls*(i-1)+sv+25:ls*(i1)+sv+32)],[u(ls*(i)+sv+1:ls*(i)+sv+8)]] vp = [vp;[v(ls*(i-1)+sv+24:-1:ls*(i -1)+sv+9)];[v(ls*(i -1)+sv+25:ls*(i1)+sv+32)],[v(ls*(i)+sv+1:ls*(i)+sv+8)]] end i=7 xp = [xp;[x(ls*(i-1)+sv+25:ls*(i-1)+s v+32)],[x(ls*(i)+sv+1:ls*(i)+sv+8)]] yp = [yp;[y(ls*(i-1)+sv+25:ls*(i-1)+s v+32)],[y(ls*(i)+sv+1:ls*(i)+sv+8)]] up = [up;[u(ls*(i-1)+sv+25:ls*(i-1)+s v+32)],[u(ls*(i)+sv+1:ls*(i)+sv+8)]] vp = [vp;[v(ls*(i-1)+sv+25:ls*(i-1)+s v+32)],[v(ls*(i)+sv+1:ls*(i)+sv+8)]] sv=ls*(i-1)+sv+1 %3rd run for i = 1:5, ls = 31 %xp1 = [xp1;[x(ls*(i-1)+sv:ls*i+sv-1)]] %yp1 = [yp1;[y(ls*(i-1)+sv:ls*i+sv-1)]] %vp1 = [vp1;[v(ls*(i-1)+sv:ls*i+sv-1)]] %up1 = [up1;[u(ls*(i-1)+sv:ls*i+sv-1)]] xp1 = [xp1;[x(ls*(i-1)+sv +22:-1:ls*(i-1)+sv+8)];[x(ls*(i-1)+sv+23:ls*(i1)+sv+30)],[x(ls*(i)+sv+1:ls*(i)+sv+7)]] yp1 = [yp1;[y(ls*(i-1)+sv +22:-1:ls*(i-1)+sv+8)];[y(ls*(i-1)+sv+23:ls*(i1)+sv+30)],[y(ls*(i)+sv+1:ls*(i)+sv+7)]] up1 = [up1;[u(ls*(i-1)+sv+22:-1:ls*(i -1)+sv+8)];[u(ls*(i-1)+sv+23:ls*(i1)+sv+30)],[u(ls*(i)+sv+1:ls*(i)+sv+7)]] vp1 = [vp1;[v(ls*(i-1)+sv +22:-1:ls*(i-1)+sv+8)];[v(ls*(i-1)+sv+23:ls*(i1)+sv+30)],[v(ls*(i)+sv+1:ls*(i)+sv+7)]]
77 Appendix A (Continued) end i=7 xp1 = [xp1;[x(ls*(i-1)+sv+22:-1:ls*(i-1)+sv+8)]] yp1 = [yp1;[y(ls*(i-1)+sv+22:-1:ls*(i-1)+sv+8)]] up1 = [up1;[u(ls*(i-1)+sv+22:-1:ls*(i-1)+sv+8)]] vp1 = [vp1;[v(ls*(i-1)+sv+22:-1:ls*(i-1)+sv+8)]] %4th run sv=ls*i+sv+6 for i = 1:15, ls = 22 xp2 = [xp2;[x(ls*(i-1)+sv+1 0:-1:ls*(i-1)+sv)];[x(ls*(i-1 )+sv+11:ls*(i-1)+sv)+21]] yp2 = [yp2;[y(ls*(i-1)+sv+1 0:-1:ls*(i-1)+sv)];[y(ls*(i-1 )+sv+11:ls*(i-1)+sv)+21]] vp2 = [vp2;[v(ls*(i-1)+sv+1 0:-1:ls*(i-1)+sv)];[v(ls*(i-1 )+sv+11:ls*(i-1)+sv)+21]] up2 = [up2;[u(ls*(i-1)+sv+10:-1:ls*(i-1)+ sv)];[u(ls*(i-1)+sv+11: ls*(i-1)+sv)+21]] end %5th run % for i = 1:3,
78 Appendix A (Continued) % ls = 24 % xp3 = [xp3;[x(ls*(i-1)+sv:ls*i+sv-1)]] % yp3 = [yp3;[y(ls*(i-1)+sv:ls*i+sv-1)]] % vp3 = [vp3;[v(ls*(i-1)+sv:ls*i+sv-1)]] % up3 = [up3;[u(ls*(i-1)+sv:ls*i+sv-1)]] % end Cp = (up.^2+vp.^2).^.5; Cp1 = (up1.^2+vp1.^2).^.5; Cp2 = (up2.^2+vp2.^2).^.5; Cp3 = (up3.^2+vp3.^2).^.5; Cp4 = (up4.^2+vp4.^2).^.5; clf hold on pcolor(xp1,yp1,log10(Cp1)) shading(type) pcolor(xp,yp,log10(Cp)) shading(type)
79 Appendix A (Continued) pcolor(xp4,yp4,log10(Cp4)) shading(type) %colorbar('horiz') %[cmin cmax] = caxis; quiver(x,y,u,v,'k') % %colorbar('horiz') % %caxis([cmin cmax]); % pcolor(xp2,yp2,log10(Cp2)) shading(type) colorbar('horiz') %caxis([cmin cmax]); %axis equal axis('tight') %[c,h]=contour(xp,yp,Cp) %clabel(c,h) %for j=1:length(x), % text(x(j),y(j),num2str(j)) %end fname = ['forcemag_',text,'_',type]; print('-dtiff', '-r600', fname)