Designing Compliant Mechanism Suspensions. Numerical Synthesis of a Monolithic Bicycle Fork

List of Figures

Figure 1: Barclays cycle superhighways London [Bre12]

Figure 2: hotchkiss suspension [Qui02]

Figure 3: snowboard cushioning after high jump[Man14]

Figure 4: compliant mechanism wheel [http6]

Figure 5: Lauf leaf spring suspension fork [Ben13]

Figure 6: advantages of the Kilo No. 1+ [hhtp9]

Figure 7: German A Force Kilo No. 1+ [http8]

Figure 8: rotational DoF bicycle fork [http11]

Figure 9: early bicycle leaf spring applications [Had14]

Figure 10: suspension variants [Had14]

Figure 11: early pneumatic leaf springs [http10]

Figure 12: two-mass suspension system

Figure 13: functional design of a bicycle suspension fork [http4]

Figure 14: structural design optimization methods [Ben03]

Figure 15: definition of the design space

Figure 16: ABQUS – ATOM Topology Optimization [Das11]

Figure 17: a) flexible segment b) rigid-link mechanism [All03]

Figure 18: FACT library case example [How13]

Figure 19: PRBM compliant mechanism synthesis example [How13]

Figure 20: Topology Optimization of Compliant Mechanisms [Ben03]

Figure 21: compliant system synthesis workflow

Figure 22: 1ststep: Finite Element Model

Figure 23: 2ndstep: Topology Optimization

Figure 24: 3rdstep: selecting and reengineering relevant topologies

Figure 25: 4thstep: sandwich design

Figure 26: neutral axis of a beam

Figure 27: 5thstep: Shape Optimization

Figure 28: 6th step: CAD modeling

Figure 29:  7th step: FE model reduction

Figure 30: 8thstep: Multi Body Simulation

Figure 31: 9thstep: prototyping

Figure 32: reversed problem formulation

Figure 33: separation of confliction design responses

Figure 34: spring elements as stiffness indicators

Figure 35: modeling technique for contradictory problems

Figure 36: design space for the topology optimization

Figure 37: expected loads

Figure 38: differing numerical results depending on the element size

Figure 39: FE mesh in ABAQUS

Figure 40: springs and boundary conditions modeled in ABAQUS

Figure 41: side face of resulting topology A

Figure 42: 3D view of resulting topology A

Figure 43: a) side face and b) 3D view of resulting topology B

Figure 44: CAD model of topology A

Figure 45: CAD model for topology B

Figure 46: Influence of width on absolute displacements and

Figure 47: Influence of width on relative displacement

Figure 48: Topology A divided into 5 sections

Figure 49: initial sandwich designs for examination of section 5

Figure 50: Influence of core thickness on and

Figure 51: Influence of fiber angle and

Figure 52: Influence of Youngs Modulus on and

Figure 53: Influence of number of layers and

Figure 54: chosen sandwich design for section 5

Figure 55: chosen sandwich design for section 2, 3 and 4

Figure 56: final sandwich design

Figure 57: 2nd iteration FE-model design space

Figure 58: result of 2nd iteration TO a) no load b) deformed

Figure 59: CAD model after 2nd iteration

Figure 60: Shape optimization of the leaf spring

Figure 61: final CAD model

Figure 62: final core body design in CAD

Figure 63: FEM simulation results a) Mises b) vertical displacement

Figure 64: MBS model of a

Figure 65: MBS results of the cyclist

Figure 66: prototype sandwich design

Figure 67: current status of the prototype

Figure 68: Influence of section 2 core thickness and

Figure 69: Influence of section 2 fiber angle on and

Figure 70: Influence of section 2 Youngs Modulus on and

Figure 71: Influence of section 2 number of layers on and

Figure 72: Dependence of and  on section 3 design

Figure 73: Dependence of and  on section 4 design

Figure 74: Influence of section 1 core thickness on and

Figure 75: Influence of Youngs Modulus on

Figure 76: Influence of section 2 fiber angle on and

Figure 77: and dz for Carbonfiber-Epoxy layers

Figure 78:  and dz for Glassfiber-Epoxy layers

Figure 79: sandwich design for a CE core

Figure 80: Mises strain for CE core sandwich design

Figure 81: weight of section 1 for CE core design

List of Tables

Table 1: disadvantages of single-piece compliance mechanisms

Table 2: advantages of single-piece compliance mechanisms

Table 3: Alternating stress on the critical nodes [Kue12]

Table 4: benchmark report of trekking bike suspension forks[Don05]

Table 5: required qualities of the bicycle fork

Table 6: springs used as stiffness indicators

Table 7: demands on the bicycle fork

Table 8: design responses

Table 9: TO design constraints

Table 10: Resulting Topologies to be proceeded with

Table 11: influence of the design parameters in section 5

Table 12: influence of the design parameters in section 2

Table 13: influence of the design parameters in section 1

Table 14: 2nd iteration TO design constraints

Table 15: compliant system quality after 2nd iteration TO

Table 16: global stiffness matrix [N/mm ; N*mm/°]

List of Abbreviations

Abstract

To satisfy modern demands in lightweight design and dynamic performance, the most advanced technologies must either support new fields of engineering science or they have to be combined with one another. One way to satisfy the high expectations in comfort and weight reduction is to implement compliant mechanisms. But the lack of examples, guidelines and resources is prohibiting a wider application. The present thesis is explaining the motivation behind the idea of designing compliant mechanism suspensions for vehicle applications, with the focus on bicycle suspension forks, considering the state of the art in design as well as the potentials of their implementation. A further look at different approaches in light weight design of bicycle suspension forks is taken. The necessary theoretical background of suspensions, structural design optimization methods and compliant mechanisms is summarized.

Finally, a guideline is developed and exemplified, which helps engineers to quickly find a compliant mechanism concept for any kind of suspension system. Numerical synthesis and iteration methods are used to significantly reduce the number of design iterations. In order to deal with multiobjective deliverables considering directional stiffness properties, the complex body structure is generated using Topology Optimization as a tool of the Finite Element Method. Every step of the design process is critically reflected by using the example of a bicycle suspension fork. The resulting bicycle fork is evaluated according to a Multi Body Simulation comparison between an existing fork and the developed compliant system. The thesis is summed up by testing a prototype. Finally, the goal of this thesis is to give the designer a rational basis for his choice of the initial form of compliant mechanisms.

1    Motivation

Limited material resources, competition in technology or a growing environmental consciousness are demanding high-performance light-weight structures at low-cost. Obviously optimal structural design is gaining more and more importance. [Hua10] The current increase of population in Megacities and the resulting decrease of space lead to sufficient discussed traffic and air pollution problems. Hence city governments try to provide incentives to ensure that more people use public transportation or vehicles with less impact on the traffic aroused problems, such as smaller vehicles or particularly pedelecs, electric vehicles or bicycles. The propagation of cleaner mobility for instance is promoted by politicians. Special bike highways, starting in the suburbs and heading towards the city centers or tax benefits for zero emission vehicles are examples for the governmental effort. In such cities that encourage commuters to use their bicycle, a continuous expansion of bicycle usage can be observed. For example in Copenhagen“… every time the city creates a new cycle track, it results in 20% more cyclists (and 10% less cars) using that stretch.” [http1] In result “52% of all Copenhageners cycle to their place of work or education every day, even when this is located outside the municipal boundary.” [http1]

Figure 1: Barclays cycle superhighways London [Bre12]

Especially in megacities like London minimal mobility is supported by establishing so called cycle superhighways (Figure 1) which allow 12,000 cyclists per hour to safe up to 29 minutes of journey time and prohibit ongoing air pollution. [Web13]

The trend in development towards smaller and more economical vehicles is demanding more consistent lightweight technology. New potentials for lightweight construction applications have to be developed for all kinds of vehicles, particularly regarding its growing importance for the range of modern electrified mobility concepts.

The example treated here is the bicycle suspension fork for commuters. Whatever the case, with or without a supporting electric engine, bicycle commuters have to cover quite long distances on a daily basis, therefore it is very important for their health and the resulting willingness to continue commuting by bike to ensure an ergonomically comfortable ride. One of the main factors is the elastic behavior of the bicycle which is prohibiting the force transmission from the street to the rider and restricts the resulting frequencies of the dynamic forces to certain ergonomically suitable areas. To enhance this factor one might consider adding more flexible suspension parts to the bicycle, which come along with certain damping behaviors. These could be for instance bigger tires or a suspension fork. Bigger tires result in a higher friction and thus more energy loss and are not preferred by riders. Most of commuters who have problems with muscular fatigue over the commuting distance decide for suspension forks. The main function of such forks are ensuring dissipation of regular harmonic excitation and preventing damage by stronger impulses, such as curbs, street holes or stones. Existing bicycle forks come along with the demand for service and maintenance and don’t necessarily satisfy the dynamic requirements due to their initial breakaway torques. Another disadvantage is the additional weight of suspension forks compared to rigid works. Accordingly a bicycle suspension fork serves as a promising main example to explain the developed synthesis approach, described in the following.

If something moving is designed, engineers classically use very stiff rigid parts connected with joints,but nature confronts us with another strategy. Almost all naturally occurring motions are realized by flexible parts which are bending. [How13] Similarly, one major approach in lightweight design is designing components for strength instead of stiffness, which results in naturally tolerated feasible displacements during strain. The following research in this thesis is taking advantage of this fact. By adding flexibility to rigid vehicle frames, a weight reduction is achieved and simultaneously suspension characteristics are added to the chassis, also referred to as compliant mechanism design. Those mechanisms exhibit major advantages compared to traditional mechanisms, but are only rarely used because of the unexplored design process and the lack of expert knowledge in compliant mechanism design. Even if many scientific papers about in-depth theory of compliant mechanisms have been published, engineers demand for inspiration by more visual and exemplary publications in the conceptual stage of the design. [How13]

2    Objective and Organization of the Research

The following thesis was born of a key idea to deliver a utilization oriented approach in compliant mechanism design. It is meant to be applied to different kinds of spring damper systems for lightweight applications and delivers the demanded applied approach of how to design compliant mechanism concepts. The main goal of this research is to develop an applicable workflow of how to design a compliant mechanism suspension concept involving multidimensional contradicting objectives. The introduced method is explained and validated using the example of a bicycle suspension fork. The advantages of such mechanisms are the reduction in the number of parts, the vulnerability, the service and maintenance and simultaneously the enlargement of the deformed volume during the suspension process. More advantages of the introduced approach in suspension system design compared to the existing ones are listed in Chapter 4.3. It is aimed at a process which allows engineers to obtain a monolithic, lightweight and maintenance free compliant mechanism which satisfies the requirements of an existing spring damper system. To achieve these high aims, modern computer simulation methods and advanced technology in material and manufacturing science is aimed to be used. If successful, the achieved methods can be developed further to obtain suspension systems or even steering through tilting mechanisms for all kinds of vehicles.

Numerical iteration methods and modern computer software is allowing us to deal with multidimensional problems and contradicting objectives. As a result, modern lightweight design is overcoming the common ways of classical engineering by questioning established directives in engineering design. Normally spring damper systems are only locally and one-directional malleable, while the highest proportion of the body mass is designed for stiffness. This proceeding is owed to the fact, that rigid parts connected by hinges are easy, and most importantly, predictable to design. In result most parts of such a system stay almost rigid during deflection and only a small part of the whole system is transforming the kinetic energy into deformation energy. If this is taken one step further, it appears to be a better idea to distribute the energy over the whole topology of a suspension system, as seen in countless examples in nature. If you allow all material to deform a little instead of maintaining rigid, designers can save weight and the quantity of parts.

So, why are current spring damper systems not just designed as big leaf springs, as for instance skateboards are? Basically those are not more than a leaf spring on wheels, but they still provide damping and energy absorption capabilities in vertical direction. At the same time it is relatively stiff for transversal and torsional loads.

In most cases the spring systems are only allowed to be flexible in one direction. The design of such systems as compliant mechanisms is involving multidimensional contradicting objectives and would have been a project of endless iterations. Modern materials with adjustable direction-dependence of strength as well as numerical software, operating with the FEM or MBS facilitate to deal with complex design problems and can most likely overcome the fear of designing complex compliant mechanisms.

Recent published papers address the issue with different approaches. In most cases the main focus is on developing new algorithms, which allow a more precise, fast or safe solution of multi objective problems. From an engineering point of view, you cannot directly benefit from such research, since it is not applicable to those design problems mentioned above. This thesis will use a simple example to develop a workflow approach for the design of compliant suspension systems by means of latest software and methods. The purpose of the investigation is to find a scalable way to design a compliant mechanism as a suspension system, combining only commercially available software and tools with a reduced number of iterations.

This first part of the thesis is explaining the motivation behind replacing spring damper systems by one-piece compliant systems. It will disclose the advantages and disadvantages of compliant systems as well as the major problems in designing such. The following part is introducing the state of the art in compliant system and bicycle fork design. Hereupon the approach for future workflows is introduced. The used software, tools and technology is established and explained. The limitations of the research are introduced, followed by a case example of a bicycle fork to illustrate the strength and feasibility of the workflow. The subsequent part is concerned with identifying the main requirements for the spring fork used as an example for compliant mechanism suspension system design as well as the identification of a feasible main load case. Finally a bicycle fork is designed, virtually tested and prototyped based on the compiled workflow. The thesis is closing with a critical evaluation of the developed method, the example and the results.

3    Limitations of the Research

Purpose of the following research is to illustrate the possibilities of numerical iteration software in engineering design. For a broad application of compliant mechanisms in suspension systems, guidelines, examples and experience is still missing. [How13] On the following pages interested users can find a brief introduction to the theory and the state of the art of compliant mechanism suspensions and subsequently an approach for the integration of numerical, iterative software in compliant mechanism synthesis. The objective is to give a feasible and scalable workflow for a fast, structured and stringent suspension system conception.

Designing a monolithic suspension system in its entirety would include detailed research and deep knowledge in all disciplines introduced within all phases of the introduced approach. To prevent going beyond the scope of this thesis, the content in Chapters 6, 7 and 8 is not aiming at developing a marketable suspension, rather than demonstrating the possibilities and strengths of this method to decrease the number of iterations and to accelerate the design process. The major advantage of the introduced method is to enable engineers to quickly create an optimal structural design for a monolithic spring damper system, without precise knowledge about the occurring load cases. More time has to be invested for each of the phases to reach a feasible quality of the designed system.

4    State of the Art

4.1     Composite Leaf Springs

As leaf springs account for 10% to 20% of the unsprung weight of vehicle suspensions, they have high potential of both reducing the gross weight and improving the riding qualities. Since composite materials have a higher elastic strain energy storage capacity as well as better strength to weight ratio compared to spring steel, they provide better qualities in supporting lateral loads, shock loads, brake torques and driving torques.[Ebk12] Leaf springs are commonly used on trucks because of their ability to carry higher loads. The picture below shows the example of leaf springs applied to a hotchkiss suspension. [Qui02]

Figure 2: hotchkiss suspension [Qui02]

During the last years several papers about composite leaf springs have been published following different focuses and approaches. Most papers objective is to draw a comparison between load carrying capacity, stiffness and weight savings of a composite leaf spring with that of conventional steel leaf springs under the same loading conditions. The results are worth a closer look, such that for instance up to“65-70% of weight reduction could be achieved by using the carbon composites”.[Dub13] Kumar and Vijayarangan carried out a design and experimental analysis of a composite multi leaf spring using glass fiber reinforced polymer. Recapitulatory composite leaf springs are found to have 67% lesser stress, 65% higher stiffness, 127% higher natural frequency and 68% less weight than existing steel leaf springs. Those results are supported by different released papers analyzing similar problems, as mentioned above and summarized in the following.

In other cases Ramakanth achieves weight savings of 67% for a simulated multi leaf spring. [Ram13] Ekbote obtains similar results. A numerically optimized mono leaf spring saves 65% weight while the maximum stress is even 30% less than that of the multi leaf steel spring. The analysis is verified by a good match with analytical and experimental results. The paper centralizes, that the FE analysis has proved to be the most convenient and simple approach to begin by for a geometry optimization of composite mono leaf springs.“The method can be applied successfully in addition to other optimization techniques.”[Ekb12]

These effects arise from the properties of composites which have “65% higher stiffness and 127% higher natural frequency” compared to steel. [Ram13]Researching the dynamic and fatigue performance of composite multi-leaf springs, it was found that both dynamic and fatigue performance are improved by replacing steel as a material for leaf springs. The fatigue life for E-glass composites is expected to be two or four times higher than that of steel multi-leaf springs. In general it was found that fatigue performance wise epoxy as a matrix material is better than vinyl ester. [Kue12] Gebremeskel designed, simulated and prototyped a single composite leaf spring for light weight vehicles. He proved that the use of composites as spring materials reduces weight, increases strength and achieves afatigue life of  cycles in his specific application. He found that his particular design is made specifically for light weight three wheeler vehicles. [Geb12]

Composite leaf springs are not only used in vehicle applications but for example also in medical applications or in sport equipment industry. In 2012 Oscar Pistorius was the first disabled person ever to compete in the Olympic Games and proved what modern materials and engineering are capable of.

Besides those tailor made applications composite leaf springs come along in all kinds of shapes in high tech sports equipment mass production. For instance snowboards, skis, longboards, pole vault poles or bicycle spring forks are using sandwich technologies to achieve the best performance properties possible for light weight structures. All of them more or less absorb shocks from the uneven surface and can withstand massive deformation and support the user in any situation in keeping control of equipment and motion at any time during high speed rides or massive jumps and impact forces. During the Olympic Games 2014 in Sochi Shaun White landed on the lip of the half pipe after flying several meters up in the air. The energy of the fall was stored inside the sandwich structure of his snowboard (Figure 3) and saved him from suffering injuries and even allowed him to win back the control over the situation. This is a great example for high performance potential of modern materials.

Figure 3: snowboard cushioning after high jump [Man14]

Regarding the current development of automotives, the usage of composites is gaining popularity. Thanks to decreasing average speed and space in the cities and due to the growing demand for more sustainable and light mobility solutions, composites are in focus of modern engineering design.

4.2     Compliant Mechanisms

“If something bends to do what it is meant to do, then it is compliant. If the flexibility that allows it to bend also helps it to accomplish something useful, then it is a compliant mechanism.”[How13]

Compliant Mechanisms can be found everywhere. Especially nature has discovered and developed compliant mechanisms over millions of years as highly effective devices to create motion between structures. Early man made inventions already included a multitude of compliant mechanisms – for instance bows, Leonardo da Vinci’s machines or steering mechanisms of the first airplanes. The need for many iterations and the complexity of compliant mechanisms made it much easier and more predictable for designers to connect rigid parts with hinges. The increasing numerical capabilities of modern computers, possibilities in using new materials and the engineering knowledge or modern design methods allow us to reconsider the implementation of compliant mechanisms. Even if compliant mechanisms have many advantages compared to traditional devices, they are just rarely used in modern engineering design. Except a multitude of scientific publications, there are not many visual, exemplary or applied resources to guide designers in the implementation of this new technology. The concept of compliant mechanisms is based on the simple fact that components can be both flexible and strong, since strength is not the same as stiffness. The widely spread misconception that something has to be stiff in order to be strong has to be overcome for the implementation of compliant mechanisms, while designing for strength instead of designing for stiffness is a commonly used procedure in lightweight design. There are four existing methods to synthesize compliant mechanisms [How13]:

•     synthesis through freedom and constraint topologies (FACT)

•     synthesis through rigid-body replacement (PRBM)

•     synthesis through use of building blocks (BBS)

•     synthesis through topology optimization (TO)

Theoretical background information about compliant mechanisms is provided in Chapter 5.4.

4.2.1     Compliant Mechanism Suspensions in Vehicle Design

Compliant Mechanisms, as explained in chapter 5.4, can be used as suspension systems. They are taking advantage of the inner damping properties and the spring characteristics of the flexible joint segments and achieve similar characteristics to spring damper systems. The theory of suspension systems is described in Chapter 5.1.

Applicated to vehicles, suspension systems mainly act as part of the wheel locating mechanism and as energy storage elements. The main functions of suspension systems are [All03]

•   Allow motion in vertical direction to separate the vehicle body from the wheels which follow the uneven road.

•   Maintain contact of the wheels with the surface.

•   Divide the body mass between all wheels.

•   Prevent the vehicle from tilting over.

Using compliant mechanisms as spring damper systems can save part count, joints and costs for manufacturing and assembly. In his masters thesis Allred has shown a compliant suspension system requires control of wheel motion in response to control forces and found that the required space and weight constraints are competing with fatigue failure as the main limiting design constrain. [All03] Irregularities of the surface over which wheels are travelling are resulting in vertical as well as horizontal acceleration. By allowing compliance between the wheel and the moving mass, energy storage elements between those two bodies store and release energy and thus are minimizing the acceleration inputs to the vehicle. The first system used on automobiles have been leaf springs, which combine the two main functions of suspension systems – energy storage and wheel location, which are separated in modern kinematic suspension mechanisms. [All03] To prevent the body mass from resonance disaster, energy is transferred into heat while storing and releasing energy due to friction. Friction appears in between oil molecules in modern dampers, in between the steel leafs of classic leaf springs or in between the layers of the core material or the fibers of composite leaf springs.

Even without kinematic joints, compliant mechanisms allow motion between its members due to deflection of flexible leafs. Compliant mechanism suspensions could for instance be adjusted to automobiles, all-terrain-vehicles, bicycles, motorcycles, snowmobiles, light vehicles or remote control cars. [All03]

4.3     Bicycle Suspension

The requirements on the comfortability, safety and controllability of bicycles in general are increasing as the quality of the technology is improving through research and development. Shock absorbers have a major effect on the performance, as they absorb vibration energy and release it in optimal time to maintain a suitable contact force between the wheel and the surface and in fact stabilize the bicycle frame. Suitable suspension systems are isolating disturbances by uneven surfaces in form of acceleration or loads within a feasible range. that is not significantly decreasing the durability of the chassis. They are furthermore keeping the contact forces between wheels and ground, to allow the rider to stabilize the ride and to stay within controllable driving conditions. [Mao09] Congruously suspension forks applicated to bicycles allow the rider to pick a straight line instead of cornering around bumps without a loss of fatigue strength, traction and speed. The difference of acceleration between the axle and the frame during an impetus or dynamic excitation is achieved by a feasibly adjusted spring damper configuration. The discussed impact event consists of a series of events. [Ore96]

1.        compression of tires

2.        compression or bending of the fork

3.        period of flight, where the tire loses contact to the surface while travelling vertically

4.        extension or back bending of the fork to the original position

5.        impact landing of the tire on the surface

The process during a bump impact leaves the frame with two acceleration peaks, one in the moment of impact and one while landing. [Ore96]

In general, three kinds of suspension systems are existing – passive, semi-active and active suspensions. Passive systems consist of a spring and a damper of constant values, where the spring stores the dynamic energy and the damper eliminates oscillations through dissipation.

Active systems consist of spring and damper as well as sensors and actuators. They allow a changing of the spring and damper values feasible to the current situation to optimize the vehicle dynamics controlled by a developed algorithm. Nevertheless, active suspension systems are not suitable for motorbikes or bicycles as their working frequency is limited to the input power and the application of complicated peripherals is expensive and needs electric energy.

Semi-active systems are capable of adjusting the damping value depending on the valve acceleration. They cannot actively control the vehicle dynamics but react suitable in a wider range of dynamic situation thanks to the acceleration dependent damping behavior. The damping coefficient is adjustable in different ways. For instance by changing the structural friction by varying the normal force between components, by manipulating the fluid viscosity by using electric or magnetic fields or by actively controlling the orifice area. Most of the existing semi-active forks manipulate the orifice area due to limitations in space, power and weight. [Mao09]

Typical bicycle suspension forks have a linear direction of travel and passive spring damper tubes and show simple kinematic behavior as explained in Chapter 5.1.In the following the most improved and promising concepts for bicycle forks, as well as traditionally existing compliant mechanism suspension forks, are introduced and explained.

Figure 4: compliant mechanism wheel [http6]

The Loopwheel, introduced in 2009, uses carbon leaf springs instead of spokes to connect the wheel hub with the wheel rim. While the tire keeps following the surface, the wheel center or hub can move relatively to the rim due to the flexible spokes and in result smoothes the excitation of the rough surface and unexpected impacts, as illustrated in Figure 4. The biggest advantage of this wheel is the small unsprung weight and the compact design, which allows the technique to be applied to very small bicycles, such as foldable bicycles. In return the travel is limited to a fraction of the radius of the rim.

Figure 5: Lauf leaf spring suspension fork [Ben13]

In 2013 Reykjavik based company Lauf introduced their leaf spring bicycle fork made for 29” mountain bikes. The fork received numerous plaudits from the industry press due to it’s unique design and the convincing weight of only 980g at a suspension travel of 60mm. Even without a damper, the fork shows exemplary maintenance and performance properties and demonstrates the excellent suitability of composite leaf springs in suspension applications. The simple functional layout is illustrated in Figure 8 above.

Figure 6: advantages of the Kilo No. 1+ [hhtp9]

The Germany based company German A offers lightweight suspension forks, such as the Kilo No. 1+ in Figure 7. While providing a travel suspension of 90mm, it weights only 1098g. Due to the smart mechanism design it comes along with some advantages such as the wheel caster, anti-drive effect, accuracy, response, or the wheel trajectory. As illustrated in Figure 6 on the left hand side, the anti-drive effect describes the behavior of the system while breaking due to the wheel trajectory illustrated on the right hand side. The curve shaped wheel trajectory prevents the fork from sinking in at the point [B] where the direction of travel is reversed at 40% of the maximum travel [C]. Also the more horizontal direction of the curve at the beginning of travel enables a more rapid response behavior. Traditionally telescopic forks describe a linear wheel trajectory and therefore don’t enable a load-dependent travel direction but in contrast the wheel caster is shortened during the fork stroke. This characteristic makes it harder to control the front wheel at suspension compression. [http9]

Figure 7: German A Force Kilo No. 1+ [http8]

Another more or less successful attempt in finding a new suspension mechanism concept is represented by the Suntour Swing Shock in Figure 8. With a weight of 1300g it offers 30mm of travel. The mechanism is mostly driven by its optical appearance and is meant to replace a rigid fork without changing the bike geometry while keeping a clean appearance of the bicycle.

Figure 8: rotational DoF bicycle fork [http11]

The first documented suspension applied to velocipedes in 1816 was a double leaf spring attached to the end of the saddle. This improvement in comfort was developed further to handlebar suspension and fist wheel suspension like constructions as depicted in Figure 9.[Had14]

Figure 9: early bicycle leaf spring applications [Had14]

In 1869 four variants of front wheel suspensions were invented by René Olivier. Steel leaf springs were placed below or above the front axle as well as a coil spring within steering head was used. The four variants are pictured in Figure 10. [Had14]

Figure 10: suspension variants [Had14]

Figure 11 pictures early suspension forks, where the whole topology functions as a leaf spring. They represent the best example for an attempt in implementing suspension forks as compliant mechanisms. No further information about the functionality, age or distribution is known.

Figure 11: early pneumatic leaf springs [http10]

5    Theoretical Background

5.1     Suspension Systems

Suspension systems in general are of major importance for the handling of the dynamic behavior of various kinds of vehicles. The main requirements suspensions are satisfying are [Gil92]:

•     separating the vehicle body from the uneven surface it is travelling on, by allowing vertical displacement of the wheels

•     dividing the vehicle mass equally on all wheels

•     prohibiting tilting over of the chassis

•     maintaining the contact to the surface at all time

•     transferring the control commands as control forces to the surface through the tires

•     smoothening of acceleration peaks transmitted from irregularities of the surface

•     damping oscillations of the system

Figure 12: two-mass suspension system

A typical suspension system is most simply modeled using two masses, two springs and one damper as illustrated in Figure 8, which represents an abstracted model of a quarter car. The sprung mass M is the main body mass of the suspension system or vehicle.represents the stiffness of the wheel,  and  represent the stiffness and damping constant of the suspension system. The damping of the tire is insignificant small compared to the damper. The natural frequency of the system is an indicator for the riding convenience and is calculated at

while the range of comfortable frequencies lies between  and  for human riders. A small unsprung mass is able to follow the irregularities of the surface better. But as  is decreased, the wheel loses the contact to the surface more easily. The frequency  which is describing the hopping motion of the wheel is an indicator for the harshness of the ride can be calculated at

Typical damping coefficients for passenger vehicles should be set up at  up to. The ratios between , ,,  and  form the characteristics or tuning of the system. [All03]

Figure 13: functional design of a bicycle suspension fork [http4]

A typical suspension spring of a bicycle is illustrated above in Figure 8. The spring is normally made of spring steel and is storing dynamic energy which then slowly released within a feasible time, dependent of the damper valve setup. The damping is achieved by a hydrostatic or aerostatic resistor which is slowing down the flow of liquids through the damper valve and as a result forms a value that is proportional to the velocity of the system. The vertical displacement of the axis and the frame of a bicycle equipped with a fork similar to that one introduced in Figure 8 with a linear and nearly vertical direction of travel can be described by the following time dependent differential equation

where  represents the relative displacement of the axle and the frame in time,  the natural frequency as mentioned above and  stands for the dynamic excitation by the roughness of the road. The dynamic behavior of a standard kinematic system, as illustrated in Figure 12, can be described by the equation above. [Dub12]

5.2     Structural Design Optimization Methods

Bendsoe and Sigmund defined structural design optimization as finding the perfect “lay-out”for a structure, which is including optimized topology, shape and sizing. These three informations address different structural design problems. Figure 14 is explaining the different categories in structural optimization. The goal of a typical sizing problem may be to perfect the cross sections of the rods of a framework, as illustrated in Figure 14 a). While satisfying constraints like equilibrium or maximum deflection other physical quantities such as mean compliance, peak stress, etc. are optimized. In sizing problems, the domain of the design model and state variables is fixed and known a priori, but the design variable is diversified. In contrast, shape optimization is aiming to find the optimum shape of a given domain, as shown in Figure 14 b). Topology Optimization on the other hand, is determining domain features like number, location, shape or connectivity of holes. (Figure 14 c) The only quantities that are necessary to perform TO are the applied loads, possible supporting conditions and the volume. Furthermore, some design restrictions like symmetry or the position of holes can be considered. The physical size, the shape and the connectivity stay unknown for the problem formulation. Below, Figure 14 shows the three principles of structural design optimization, as explained above.

a)        Sizing Optimization of a truss structure

b)        Shape Optimization

c)        Topology Optimization

Figure 14: structural design optimization methods [Ben03]

The information gathered from all three principles describe the optimal lay out of the structure. Those principles are not represented by usual parametric functions but by a set of distributed functions. A proper design formulation for TO is achieved by a suitable parametrization off the stiffness tensor of the continuum which is represented by that very set of distributed functions. The method used in this thesis is a formulation for minimum compliance, which is equivalent to maximum global stiffness, as described in the following in Chapter 5.2.1.

5.2.1     Topology Optimization

In general, TO problems can be formulated in two ways, which both result in equivalent topology solutions. They can either be designed for finding the minimum volume of the structure constrained by a certain compliance condition, or for a minimized compliance limited by a maximum volume constraint. [Gha09]

In practice, structural optimization techniques are identifying the best performance for a structure limited by various constraints. Those constraints could for instance be an amount of material as well as maximum volume, displacements or forces. [Hua10]

Using topology optimization, engineers can find the best concept design that meets the design requirements. Topology optimization has been implemented through the use of finite element methods for the analysis, and optimization techniques based on the method of moving asymptotes, genetic algorithms, optimality criteria method, level sets, and topological derivatives. According to Bendsøe, it is aspired to explore the topology field with all its possible results related to the given requirements, loads, support conditions and volume. To achieve this, different strategies are pursued varying the design responses, their weighting methods and the strategy of the problem formulation. The most simple way is to design the numerical problem formulation for minimum compliance or maximum global stiffness. A general method for a design of the optimal shape is described as a guidance of how to distribute a given amount of material, or incremental mechanical elements, inside a given empty volume, the design space – the so-called material distribution problem. Those elements can be considered as incremental bodies occupying a domain  inside the design space , with  or . The design space can also be called the reference domain or ground structure and is formed by the volume and shape where material distribution can is allowed and by the defined positions of applied loads and boundary conditions as illustrated in Figure 15. [Ben03]

Figure 15: definition of the design space

The problem of where to put material and where to put holes inside the allowed area can be described mathematically as how to find the best variable stiffness tensor  for maximum global stiffness . Therefore some values are introduced:

•     the equilibrium

•     the arbitrary virtual displacement

•     the kinematically admissible displacement fields

•     the body forces

•     the volume

•     the boundary tractions

•     the traction part

•     the set of admissible stiffness tensors

•     the limit of the resource

•     the linearized strains

The energy bilinear form (the internal virtual work of an elastic body at the equilibrium for a virtual displacement)

combined with the load linear form

form the above mentioned formulation for minimum compliance

If the same finite element mesh is used for both of the relevant fields, displacement  and the locally constant stiffness , we can reformulate the problem formulation for global stiffness as

using the global stiffness matrix , consisting of the element stiffness matrices

The admissible stiffness tensors  can be defined in various different ways as explained in the following. Since the goal of TO can be described as finding the optimal placement of a given isotropic material in space, which is similar to a black and white image,some pixels (or voxels in three dimensional terms) contain material (black) and some remain void or empty (white). By discretizing the volume into a finite element mesh, the TO should lead to a black –white raster representation of the optimal structure which is represented by the optimal subset  of material points or elements as part of the reference domain . This leads to the hereby used tensor formulation limited by the maximum volume . [Ben03]

This in turn reasons, why minimum compliance design is to be formulated for a fixed maximum volume to result in a well-defined 0-1-structure. Therefore, to solve this discrete valued design problem (0-1-problem) a new approach is needed. Therefore it is most common to replace the integer variables with continuous variables and later on transform the solution to discrete 0-1-values by introducing some form of penalty. This method can described as a density problem formulation, since each element can not only be completely filled with material or not filled up with material at all, but can also contain some theoretical material which has a density in between the chosen material and zero. This leads to a formulation where the stiffness matrix is a function of another function, which can be interpreted as the “density of material” in each element, since the volume of the structure can be illustrated by .  in this case stands for the material properties of the chosen isotropic material, which means that the density of each element can be interpolated between 0 and . This SIMP-method (Solid Isotropic Material with Penalization) is most commonly used in TO problem formulations. A penalization factor is introduced, which is also called power. [Ben03]

Using the SIMP method  is chosen, which automatically results in a black-and-white design, because the benefit of an element which is not completely filled with material is economically small compared to the cost of its volume. Elements with intermediate densities are avoided for an optimal design. Usually  is to be chosen for proper optimization results. One can even discover the existence of a global optimum for discrete minimum compliance formulations for a large value for.

Since numerical solving methods include interpolations, the results are never completely free of “grey”, neither black nor white, material. This fact raises the question, if the SIMP-modal can be interpreted physically and the doubt, if the result is of physical relevance, because usually there is no material existing, which satisfies the properties of all elements of the resulting topology. Considering modern numerical material and solver methods, Bendsøe summarizes, that indeed the SIMP model is satisfying the requirements for a physically relevant method, if for the Poisson Ratio  of the base material with stiffness tensor  the power  appliesto [Ben03]:

Nevertheless the interpolation scheme using the integer format doesn’t solve the problem of non-existing solution to the distributed problem and demands for further consideration due to the goal of a well-posed distributed design problem. [Ben03]

5.2.2     Topology Optimization of Compliant Mechanisms

Using Topology Optimization methods to synthesize compliant mechanisms is a process which derived from the design of Micro Electric Mechanical Systems, which cannot be assembled in the usual way, using for instance hinges, joints or bearings. The main task in rigid-body mechanism synthesis is to design the ratios between output and input displacements and/ or forces as well as to control the output path and amplification of the displacement. It is important to use geometric non-linear element formulations, because compliant mechanism synthesis is based on large displacements theory. [Ben03]

The goal of the synthesis procedure is to maximize the desired output forces/ displacements for given input forces/ displacements. A possible optimization problem formulation for  is introduced in the following

Using the vector , which is representing the degrees of freedom filled with either  or , the displacement at the output point can be expressed at

where the sensitivity of the displacement is given at

This compliant mechanism optimization problem is similar to the compliance minimization problem explained in Chapter 5.2.1. A single objective function is to be minimized while constrained to a minimal volume. [Ben03]

5.2.3     Multiobjective Optimization

To obtain stiff structures, strain energy has to be minimized. Since the strain energy for the whole volume is minimized, the conflicting requirements for a partial flexible but stiff structure are forming a contradictory problem that cannot be solved uniquely. The main purpose here is to find the right trade-off between vertical stiffness and horizontal flexibility using the concept of Multiobjective Optimization, which is a method of how to optimize several functions while additionally satisfying several constraints.

The n-dimensional solution vector  is bounded by  with  for each decision variable. All bounds form the decision (variable) space , where the objective m-dimensional function vector  defines a mapping from  to the objective space . Additionally the problem can be constrained by and . [Due09]

This formulation is representing a challenge of comparison, where different solutions have to be compared and result in the hopefully most desired trade-off between different competing solutions. The dominance relation between the used decision variables is to be set up and thus represented by weighting factors, which control the importance of each objective function related to each of the others. [Due09]

Defining the importance of different requirements can be done in various ways. The Multiobjective Topology Optimization function can roughly be defined by , minimizing the sum of different optimization functions  for a number of  objectives, in which the importance of achieving the required goals for each objective is defined by the specific weighting factor . Since the formulated problem is including many conflicting objectives, the results will always be a trade-off, where the algorithm tries to find the optimum between all involved requirements based on the weighting factors while satisfying the constrains.

5.2.4     ATOM – Topology Optimization using ABAQUS

Figure 16: ABQUS – ATOM Topology Optimization [Das11]

The simplest type of Topology Optimization is designing for minimum compliance, to obtain the lightest possible design for a certain given stiffness or achieving a maximum global stiffness with a given amount of material. The newly introduced module ATOM in version 6.11 features Topology and Shape Optimization and is capable of dealing with parts and assemblies, large deformations, contact, non-linear materials, manufacturing restrictions and offers a export function to CAD. The recommended workflow using ABAQUS ATOM to design optimized structures is pictured in Figure 16.

ABAQUS is achieving the required lay-out optimization by scaling the relative densities of elements within a given design domain. Intermediate density elements are not interpreted but penalized and not favorable in the final solution. The formulated objective function is always a weighted sum of the multiple design responses, while reference values are subtracted from the design responses as constants and aren’t taken into account as objectives. Additional constraints are derived from geometrical restrictions such as frozen areas, symmetry or demold control. While performing the Topology Optimization task, optimization output is generated for each optimization iteration step and can be tracked as history. The resulting design surface can be shown depending on the chosen cut based material fraction, or ISO value. [Das11]

5.2.5     Shape Optimization

SO typically is carried out additionally to optimize the structure derived from a previously performed TO. In general,shapes are defined by the oriented boundary surfaces of the body. SO computes the optimal form of these boundaries. By means of Finite Element Analysis this means, that the number of elements and nodes, as well as the connection between them, is given. During the SO process nodes are moved slightly in order to maintain a uniform stress distribution. The part count of members and holes is not changed during the analysis. [Ben03] This is illustrated above in Figure 14.

5.3     Leaf Springs

The main function of leaf springs in a spring damper system is to store strain energy dissipating from impacts, vibration or deformation and slowly release it under damped conditions. Stored elastic strain energy  for a uniform width can be described by the maximum tolerated stress , the Young’s Modulus  and the density  as follows [Kue12]

The capacity to absorb potential energy during deformation is called resilience. Optimizing leaf springs of varying width result in triangular shaped geometry, because the bending stress then is  uniform at any cross section and the spring has a higher resilience. [Kha15]

In general, composite fibers have a good ability of storing strain energy in fiber direction. [Kue12] Leaf springs function as linkages for holding the wheel in position and consequently require a high stiffness to position the wheel.

5.4     Compliant Mechanisms

Compliant Mechanism are one-piece components which transform an input force or displacement into a desired motion, force or deformation and represent assemblies from rigid links, pins or slider joints and springs, as illustrated in Figure 17and furthermore described in Chapter 4.2. At least a portion of its motion is gained from the deflection of flexible members, as illustrated by the examples in Figure 20. Deformation energy is stored in the system and thus stiffness and damping can be included in the mechanism, allowing suspension properties along a prescribed path. Several papers have been published trying to replace spring damper systems with compliant mechanisms, including automotive suspensions, industrial machinery or mountain bike suspensions. [All03]

Figure 17: a) flexible segment b) rigid-link mechanism [All03]

The following tables summarize the advantages (Table 2) and disadvantages (Table 1) of compliant mechanisms compared to rigid-link mechanisms.

Table 1: disadvantages of single-piece compliance mechanisms

Table 2: advantages of single-piece compliance mechanisms

5.4.1     Synthesis through Freedom and Constraint Topologies

To face the challenge of how to constraint a rigid body in a way it meets all required DOFs, this method is introducing a library of geometric shapes in combination with a systematical framework and process. The library is based on the screw theory and contains all relevant information about the kinematic, elastomechanic and dynamic properties of those shapes. According to the screw theory, an infinitesimal motion can be represented by a 6 DOF vector, visualized as a line along and about which rotation and translation is allowed. FACT is generally used to rapidly generate and visualize complex compliant concepts rather than mathematically detailed. The FACT method is mostly used for approximated infinitesimal motions in nanoscale applications. Figure 18 illustrates the use of the library by a case example. [How13]

Figure 18: FACT library case example [How13]

5.4.2     Synthesis through Rigid Body Displacement

Mechanisms are commonly seen as motion or force transferring mechanical devices, assembled of rigid links and movable joints. The PRBM method is taking advantage of the common understanding in rigid-body mechanism analysis by simply replacing joints with equivalent compliant members. In fact, PRBM will not identify new mechanism topologies, but it is a convenient way to transform already existing or easy to handle rigid body mechanisms into compliant mechanisms. To perform PRBM one can follow the following four steps:

1.        Identify the rigid-body model

2.        Replace rigid links and/ or joints with equivalent members

3.        Develop the pseudo-rigid-body model

4.        Select material and size of the compliant members

5.         

A case example for the rigid-body replacement method is illustrated in Figure 19. [How13]

Figure 19: PRBM compliant mechanism synthesis example [How13]

5.4.3     Synthesis through use of Building Blocks

The BBS method basically starts with the decomposition of design problems into subproblems, which are separately solved by implementing standardized components. Given that the designer is familiar with the behavior of the system and the available building blocks, the overall function can be broken down to subsystems with specific functions. The process relies on the intuitive decomposition of the design engineer and the valuation of the multiple solutions for each subproblem. BBS is based on three major parts:

1.        library of building blocks

2.        functional behavior library of building blocks

3.        means of functional decomposition

5.4.4     Synthesis through Topology Optimization

Topology optimization techniques are best used for the creation of novel compliant mechanism design solutions, which cannot be identified by the other synthesis methods. As the focus in this research lies on the synthesis through TO, the method is explained above throughout this thesis. Examples for compliant mechanism application in vehicle design are given in Chapter 4.2.1, while the synthesis of compliant mechanism suspensions through TO is introduced more detailed in Chapter5.2.2. Structural design optimization methods in general are explained in Chapter 5.2, containing theoretical background of TO, TO of compliant mechanisms, multiobjective optimization and the application of TO in the commercially available software ABAQUS.

Until now Compliant Mechanisms are only used commercially in micro technology applications such as micro grippers, which allow the implementation of minimalistic mechanisms, which can’t be assembled from links and joints. Figure 20 illustrates the problem formulation and the results for the two dimensional designs of a) a force inverting compliant mechanism and b) a compliant gripping mechanism.

Figure 20: Topology Optimization of Compliant Mechanisms [Ben03]

By the example of a mountain bike rear suspension Allison developed a generalized method for the optimization of a compliant suspension topology. The complicated assembly of rigid links and the air shock is found to be expensive and heavy. The rear triangle is to be replaced with a monolithic compliant system. He used simplified assumptions and concentrated on path accuracy, system mass and longitudinal rigidity in order to develop a basic, scalable method. He took aesthetics and ergonomics into consideration, while comparing different optimization results depending on different statistical selection methods. [All12] His method is only confined to rigid link compliant systems and assumes that the designing engineer is capable of superpositioning the topologies derived from the independent load cases. Also it is only applicable to two-dimensional problem formulations. (Chapter 5.4)

Tromme developed a method to design the compliant suspension fully based on Topology Optimization for all the different criteria. He managed to design compliant suspensions only using automated processes to deal with the biggest challenge of Compliant Mechanisms which is understanding both mechanism analysis methods and the deflection of flexible members but also the interaction of the two methods in a complex system. His goal was to define a robust end efficient method for how to design a compliant suspension fully based on a two dimensional topology optimization for all different criteria. Before, the complexity of the elastic behavior of Compliant Mechanisms could only be dealt with by various iterations. Due to the multiobjective formulations of criteria for vehicle suspensions, for the stroke length, rigidity of the suspension, roll center height or bounce and roll movement, he divided the design process into five steps. First he designed for flexibility using a mutual mean compliance formulation, to maximize the displacement in the direction of the shock. In a second step he designed for rigidity using a compliance formulation, such that the mechanism is stiff enough to support different loads and reactions, to transmit the wheel travel into the shock absorber and support the reaction loads. Step three contains the design for lateral rigidity to obtain a mechanism which is stiff enough to support lateral loadings, which resulted in similar topologies to step two. In step four he limited the variations of camber angle and track width due to bounce movement of the vehicle and found that the bounce movement criteria is already satisfied by the mechanism. In a final step he reduced the variation of the camber angle and the track width due to roll movement and found that it is impossible to satisfy the other criteria when roll criteria was introduced. He solved this problem by using relaxation factors which allow the violation of different constraints for each numerical iteration, when there is no solution. [Trom11] Even though this first approach in fully automated, numerical synthesis of compliant mechanism suspensions resulted in an applicable topology the chosen solution is not very clear due to a not stringent combination of the obtained solutions of all five steps. Also the usage of non-commercial topology optimization software is not useful for economic engineering design, because it is not available, it only works for two dimensional problems and in this case it only supports linear finite element formulations which result in bad solutions for large deflections. This thesis however, offers an illustrated guideline of how to use commercially available software for the synthesis of compliant mechanism concepts, without the need of subsequent superposition of topologies optimized for contradictory load cases.

5.5     Composites as leaf spring material

The purpose of this Chapter is to give a rough insight into theory and application of composites and sandwich design in order to understand how to choose the right materials and which material performance limits are most important to meet a feasible fatigue life.

E-Glass fibers offer a good balance of mechanical, chemical and electrical properties. It is most likely combined with epoxy matrix material, which offers high strength, low shrinkage, good adhesive properties, chemical resistance, low cost and low toxicity. [Ebk12]

E-Glass/Epoxy should be chosen for composite leaf spring material. Epoxy is the better choice of matrix resin for composites, even if vinylester is cheaper. Composite leaf springs get a fatigue life of approximately twice as long as conventional steel springs. The ultimate failure for epoxy occurs not under the strain of 7%, while for vinyl ester it is only 4,5%. In return one has to consider a safety factor of 3,14 for E-Glass/Epoxy, while for steel springs a safety factor of only 1,24 is needed. Kueh and Faris investigated the static and fatigue behavior of composite and steel leaf springs. They showed that the fiber orientation had no significant influence on the fatigue life of the material. Logically the stiffness and strength design of the leaf spring can freely be lay-outed by means of varying fiber directions, without affecting the durability. [Kue12] Also Gebremeskel discovered the advantageous durability properties of epoxy as matrix resin even for flexural failure. [Geb12]

Table 3: Alternating stress on the critical nodes [Kue12]

The most suitable element type for modeling composite surfaces in FE is found to be SHELL 99, which is an 8-node three dimensional element with six degrees of freedom at each node. Gebremeskel also recommended a manufacturing system for leaf springs using plywould as the mould material. The glass fibers should be cut to the required length, before they can be deposited on the mould layer by layer. One should wait for 5 or 10 minutes after draping a layer for the fibers to soak the epoxy resin solution. Then remove the spring from the mould. [Geb12]

6    Synthesis Approach

The main goal of Topology Optimization calculations is to avoid endless design iterations to save time and money and concomitant represents a method to fulfill the design requirements in the most optimal and feasible way. The structure of a vehicle suspension is required to be stiff in driving and transversal direction, but allows the structure to deform when strained by loads in vertical direction. The strain energy as a stiffness representing objective should be maximized in order to increase the transversal stiffness of the structure. At the same time the kinetic energy of the whole system is reduced by energy conversion into strain energy. In order to achieve a proper suspending behavior, the strain energy has to be maximized during vertical wheel travel. Finding a consensus between those conflicting requirements and their implementation in the optimization function is the central issue in the following chapter.

To solve this problem this paper claims to execute a full simulation considering all load cases and constraints, which allows a more vague and conceptual starting point of the design process and renders final subjective superposition superfluous. The problem here is that orthotropic material behavior and the advantage of variable layer setups, fiber directions and material mixes cannot be considered within the Topology Optimization of commercial software yet, since the direction of the fibers is dependent on the resulting topology and vice-versa, so it is known initially. The proposed solution to this difficulty is a numerical and iterative combination of different requirements and is described in two steps. Chapter 6.1 is introducing targeted measures to synthesize a compliant mechanism compliant system in a flow chart format. To deal with the contradicting problem of the directional stiffness formulation, a modeling technique to translate those objectives into design responses is described in Chapter 6.2.

6.1     Workflow

It is proposed to divide the design process up into certain steps, as illustrated in Figure 21. The phases of the proposed workflow are explained hereafter. The pictured guideline is not entirely stringent, but demands for a reduced number of iterations for smaller sections of the design space, depending on the reliability of the input values in the beginning. In return it allows to start with estimated values, if the accurate load cases are not known.

Figure 21: compliant system synthesis workflow

1st step: Finite Element Model

Figure 22: 1st step: Finite Element Model

The very first step would be to model a dependable FE model of the design space, ascertaining the minimum element size and detecting all significant influence coefficients on the resulting topology. First of all the simulation is executed using isotropic material and assuming the identified or assumed worst case scenario. The goal is to synthesize a conceptual design of the optimal structure to proceed with in the next steps. The FE model should consist of quadratic and not reduced elements.

2nd step: Topology Optimization

Figure 23: 2ndstep: Topology Optimization

During the second step,the TO field is explored by diversifying constraints, design variables and solver settings. Not one but several possible designs can result from this variation and allow a deeper comprehension of how specific settings influence the different parts of the resulting topology and how those finally affect the behavior of the system.

3rd step: Selecting and reengineering relevant Topologies

Figure 24: 3rdstep: selecting and reengineering relevant topologies

For the third step, the comprehension of the load-sensitive topologies is necessary to verify the results and to choose the relevant topologies out of the appearing compliant system designs, which should be done considering for instance manufacturing requirements, costs, optical appearance or reliability. After exploring the topology field,the selected topologies are remodeled and simplified using CAD Software.

4th step: Sandwich Design

Figure 25: 4th step: sandwich design

The fourth step is involving several simulations including different isotropic materials such as glass fiber or carbon fiber in combination with epoxy. The optimal combination of values for a chosen number of geometrical variables is examined cautiously approaching the optimal design using systematically designed simulation experiments. A limited number of simulations is carried out, to allow the comparison between the quality of different topologies under the influence of varying the core layer thickness, the beam width or the arrangement of the layers and materials. The structures which exhibit the most convenient flexibility to stiffness ratio are chosen to be proceeded with.

Figure 26: neutral axis of a beam

One advantage of using fiber reinforced plastic composites is the possibility to artificially apply directed stiffness without significantly effecting flexibility in another direction by setting stiff layers aside for the neutral axis of a beam as shown in Figure 26. This step is resulting in an optimal sandwich structure for each of the chosen topologies. Iterations only have to be carried out within these topologies to optimize stiffness or the smooth division of loads using further TO for smaller sections of the global design space.

5th step: Shape Optimization

Figure 27: 5th step: Shape Optimization

Within the fifth step Shape Optimization is carried out to flatten stress peaks in certain areas. After finishing the design of the diameter using for instance sandwich structures, a better load contribution can be obtained using Shape Optimization. Due to the disability of considering displacements as constraints this step will result in a decreasing flexibility and is therefore counterproductive regarding the need for a maximum vertical flexibility.

6th step: CAD Model Finishing

Figure 28: 6th step: CAD modeling

In step six the most promising structural designs are remodeled for further applications using CAD software. It is aimed at a model as precise as possible reproducing the design taken from the Shape Optimization. After finishing the CAD modeling, slight changes should be made to the sandwich design in order to meet the required suspension behavior.

7th step: FE Model Reduction

Figure 29:  7th step: FE model reduction

In step seven the FEM calculations are used to prepare the remaining designs for their application to a the MBS environment. As a result each compliant mechanism is reduced to a point mass and direction-dependent spring and damping characteristics matrix between the joints. These values allow an implementation of the one-piece spring damper system into a MBS environment.

8th step: Multi Body Simulation

Figure 30: 8thstep: Multi Body Simulation

Within the eighth step it is furthermore aimed at selecting of the most adequate structure for realistic calculations of the dynamic behavior of the fork, assembled to a MBS vehicle model under realistic driving conditions on different surfaces. The best design out of the remaining ones from the last step is chosen to be proceeded with. The forces or loads affecting the vehicle in motion and the design of the compliant mechanism act in mutual interdependence. This may imply the necessity to carry out further iteration loops, improving the component from the 1st or the 4th step, before progressing to the prototyping phase. Depending on the deviations of the assumed forces from the resulting forces in MBS, a second iteration of the topology starting with the basic design space model has to be carried out. If the simulated forces are consistent with the assumed forces, but the behavior under realistic conditions doesn’t satisfy the favored requirements, a sandwich re-design should be carried out, maybe followed by a eigenfrequency driven shape optimization.

9th step: Prototyping

Figure 31: 9th step: prototyping

Finally the existing spring damper system is replaced by a functional model of the compliant system and tests can be carried out under real life conditions.

6.2     Modelingcontradicting Formulationsasviable Design Responses

The desired properties of the replaced spring damper system are resulting in conflicting demands for the optimization task respective the strain energy optimization function. Basically those can be broken down to a one directional required flexibility and the demanded stiffness in the other two directions. In simple terms, the task is to maximize the travel in load or z-direction, while minimizing the displacement in x-, y- and torsional direction as well as the volume for a given load case. In typical TO stiffness is determined by maximizing the strain energy for a given reduction of the volume. The main task in numerical compliant system synthesis is to solve the insoluble problem of the conflict between maximizing and minimizing strain energy within one optimization task, as derived from the theory of Topology Optimization in Chapter 5.2. This thesis is proposing a conversion of stiffness related requirements into other non contradictory constraints.

In conclusion, again the main goal of the executed numerical optimization process is to obtain a high flexibility in the travel direction, so the objective function is to be designed for maximum flexibility under transversal and volume constrains.

Flexibility in travel direction can be derived from the simulation by reversing the problem formulation of the optimization task. A required displacement for a given force is equivalent to a forced displacement and a minimal resulting force. In this approach z-directional flexibility is achieved by forcing the design spaceto deform in travel direction as a load case and simultaneously designing the objective function for minimized strain energy in the whole volume,while minimizing the stiffness in travel direction. Flexibility in travel direction is achieved by minimizing the resulting unidirectional force at the clamped support.

Previous examinations showed that forces are no usable design responses, neither for the objective function nor as constraints. In order to obtain indicators for stiffness or flexibility as design responses, it is recommended to use spring elements in conjunction with their displacement as design responses. The developed method of translating load case scenarios into compatible design responses is introduced as listed below and explained separately in the following.

·        conflicting optimization tasks are separated by reversing the problem formulation (Figure 32)

·        flexibility in travel direction is achieved by applying forced travel as a load case and simultaneously minimizing the reaction force, volume and strain as optimization functions (Figure 33)

·        stiffness in the transversal directions is achieved by using springs as force or stiffness indicators and limiting their displacement as optimization constrains (Figure 34)

Figure 32: reversed problem formulation

The basic problem in numerical compliant system synthesis is specified in Figure 32. The goal is to achieve a topology which is deformable in z-direction. For an ideal structure the area A’ of the design space is displaced by the maximum travel distance under the maximum load, while maintaining an acceptable stiffness in x- and y-direction. These requirements can be modeled as illustrated in Figure 33. The design area is strained by a forced displacement in z-direction and loads in the transversal directions x and y. Hence a reaction force  and the displacements  and are measurable results that all are minimized factors of the optimization function and that are not contradictory formulated respective minimization of strain energy and volume.

Figure 33: separation of confliction design responses

Since the reaction force as a design response doesn’t lead to satisfying results and the algorithm is not capable of handling competitively formulated multiobjective optimizations,an adequate modeling technique is proposed in Figure 34. If the transversal force  is replaced by a spring element defined by the spring constant  the force can be applied as a forced displacement of the spring element.can be defined freely at .

Figure 34: spring elements as stiffness indicators

The proposed solution from Figure 33 to the problem illustrated in Figure 32 is realized by using spring elements as stiffness indicators, as explained in Figure 34. In combination the developed modeling technique of the described contradictory problem is illustrated in Figure 35.

Figure 35: modeling technique for contradictory problems

7    A Bicycle Fork as a Case Example

Since the main focus of this thesis is not to develop a perfectly engineered and marketable compliant mechanism bicycle fork, the required qualities and properties are based on thoroughly researched values, determined by benchmarking and reference test methods.

7.1     Design Space

Figure 36: design space for the topology optimization

The design space defines the volume in which structure is allowed to exist without interfering with other elements. Due to the limited space in the crown region of the bicycle fork (Figure 36: dark), the design space is separated into two regions which will be optimized separately. The suspension area, pictured in the right picture in Figure 36as the lower and brighter part,marks the space, where a flexible structure is pursued and represents the main optimization problem in this approach. The crown region will not take part in the suspension procedure, but will transfer the forces to the headtube or steerer. The vertical distance between crown and front hub is 450mm, to simplify modifications from rigid to suspension forks. The outer dimensions were determined by the maximum dimensions of a 28’’ bicycle tire, the hub, the distance between the pedals and the bicycle frame.

7.2     Benchmarking

Hence to non existing values of occurring loads while riding a bicycle the required stiffness is determined by analyzing existing forks of established brands. Using an existing test report about various leading Trekking Bike suspension forks the benchmark report in Table 4 was formed.

Table 4: benchmark report of trekking bike suspension forks[Don05]

The requirements of the bicycle fork to be developed are presented in Table 5. To benefit from the extensive experience in suspension design of established companies, the qualities of the here new developed example are mainly gathered from benchmarking and standard test norms for bicycle parts.

Table 5: required qualities of the bicycle fork

7.3     Load Cases

In bicycle literature reliable load cases for bicycle design are missing. They are either integrated in-house knowledge which relies on experience or on extensive experimental data or exist as imprecise empirical values without any prove of accuracy. Therefore the load cases in this case example are defined by benchmarking and norms for bicycle testing. Figure 37 illustrates all potential loads factored in by the following examinations. All applied forces are derived from existing standards or other similar sources below.

Figure 37: expected loads

The pictured loads in Figure 37 are calculated as follows.

Torsional Stiffness

The torsional stiffness, represented by  is calculated from the benchmarking in Table 4, which is . Relating to a chosen displacement of and regarding a paving width of standard bicycles of, the required load can be calculated at

That means that a displacement of  is tolerated at an applied force of 370 N.

Braking Stiffness

The so called breaking stiffness is represented by the load  plus the length  and is calculated at

Transversal Stiffness

The transversal load is calculated at the worst case scenario – riding on a rough road. The maximum transversal force can be calculated using the formula . The friction coefficient of a pneumatic tire on rolled crushed rock is. The force of gravity can be assumed as , estimating a total weight of rider and bicycle of  and a weight distribution between rear and front wheel at . In that case the transversal force results in

For this use case a maximum deformation of  shall not be exceeded.

Vertical Stiffness

The maximal load in vertical direction at the required travel of  is derived from benchmarking at [Don05]

8    Synthesis of the Compliant System Bicycle Fork

8.1     Finite Element Model

Preparing the FE model contains partitioning the design space to obtain a smooth mesh, constraining different parts and distributing the stresses at transmission points on bigger areas and finally applying loads and fixations before setting up the TO in the following step. The elements should not be too small, due to the time consumption in simulation, but also should be small enough to provide tolerable differences in the results. Therefore certain simulations with the same settings but decreasing element size are performed. It is found that an approximate element size of 12 mm leads to a deviation of less than 1 % compared to the results with a 3 mm Element size meshed model by acceptable computing times.

Figure 38: differing numerical results depending on the element size

Figure 39: FE mesh in ABAQUS

8.1.1     Applying Loads and Boundary Conditions

The key focus of this approach is to obtain a structure which is stiff in the horizontal directions and flexible in vertical directions. Integrated into the Optimization Function this means maximizing the strain in x- and y-direction while minimizing it in z-direction, which is not possible. To obtain a robust Optimization Function we need to translate those conflicting optimization demands, as described in Chapter 6.2. The FE model is pictured below in Figure 40. It is resulting from the modeled springs as well as the applied loads, boundaries and displacements, summarized in Table 6 and Table 7 and explained in the following.

Figure 40: springs and boundary conditions modeled in ABAQUS

Five springs are used to control the directional stiffness of the resulting topology as indicated in Table 6. Each of them is connecting two points, which names can be found in Figure 40.

Table 6: springs used as stiffness indicators

Furthermore the loads and boundary conditions are modeled as described inTable 7. The torsional moment is named “torsion” and the rotational boundaries “headtube_fix”, which additionally prevent the whole structure from tipping and thereby ensure the load driven design of the resulting topology. The braking force is represented by the displacements “x” and “x_wheel” of the springs “x” and “x_wheel”. The transversal force is equally modeled by the displacements “y” and “y_wheel” of the springs “y” and “y_wheel”. The required vertical flexibility is obtained by using the  force equivalent stiffness of the spring “z” and the forced displacement “wheel” as well as with the boundary “z”.

Table 7: demands on the bicycle fork

8.2     Topology Optimization

8.2.1     The Optimization Task

The optimization algorithm consists of a sum of different optimization functions for each design response multiplied by a weighting factor . The design responses used in this TO are listed below in Table 8. Additionally z-x-planar symmetry is applied for the whole topology. This has a stabilizing effect on the asymmetric loaded model.

Table 8: design responses

An optimization function considering all available design responses can be described by the following formula, as described in Chapter 5.2.1.

In addition the Element type can be either quadratic or linear, whereupon non-linear analysis is recommended for large deflections. Planar symmetry between both sides is applied as a manufacturing constrain. Also reduced element formulation is to be avoided.

The optimization function finally is formed by minimizing both equally weighted design responses and . Optimization constraints are applied as described inTable 9. A tolerated displacement of the wheel axle connection point of  on each side due to the torsional load is leading to the torsional constraint. The displacements of the spring fixation points of respectively are reasoned in Chapter 6.2.

Table 9: TO design constraints

8.2.2     Topology Optimization

The performed TO task mainly resulted in two different topologies reliant on the degree of volume reduction. In the following it is first proceeded with those two models,illustrated both in side face and in a three dimensional view in Figure 41,Figure 42 and Figure 43. The constrained volume reduction and the ISO value used for the exported figure are summarized in Table 10.

Table 10: Resulting Topologies to be proceeded with

The first identified topology has a rather two dimensional shape, as deformation is taking place essentially in the x-z-plane. It shows promising transversal stiffness properties and offers an excellent outset position for further research. Since the results for a volume restricted to a fraction of the initial value ,  or  all result in a similar topology, the method introduced in Chapter 6.2 can be regarded as robust. The differing topology obtained by a volume restricted to  marks the Minimum limit of volume restriction and is later on found to be insufficient to meet the demands.

Figure 41: side face of resulting topology A

Figure 42: 3D view of resulting topology A

Figure 43: a) side face and b) 3D view of resulting topology B

8.3     Selecting and Reengineering relevant Topologies

The two identified relevant topologies were exported into CAD and simplified into examinable components. Due to the fact, that topology A is only deformed in the z-x-plane it is resulting in a two dimensional extruded object, which allows to proceed with the design of the sandwich material, layer and stack up properties. Since it is trivial, that a wider topology is resulting in a better transversal stiffness and an optimized transversal stiffness to vertical flexibility ratio, the composite leaf spring is designed as wide as possible, regarding only esthetic aspects and considering aerodynamics. This fact is explained below in Figure 47 on page 60. The resulting spring component with a width of  is illustrated in Figure 44. In Chapter 8.4 the number of layers, material types and fiber directions are varied for all members of the topology, to identify the optimal sandwich design for the desired requirements.

Figure 44: CAD model of topology A

Unlike topology A, the second determined topology B has a three dimensional topology and offers an additional unknown, the angle of the neutral plane for vertical displacement. The optimal solution is also described in the next step in Chapter 8.4.

Figure 45: CAD model for topology B

The topology B in the picture above is found to be negligible after first FE simulations, because its structure is way too flexible to meet the required transversal stiffness criteria.

8.4     Sandwich Design

In this fourth step, the material and sandwich structures of the obtained topologies are optimized and finally the chosen topologies are compared. For a better comparability, parameters are introduced, which represent the ability of the structure to offer vertical flexibility while being as stiff as possible in transversal and driving direction. It can be examined, that the most demanding task in designing the spring fork is to keep the required transversal stiffness for the needed vertical deformation in case of an impact. Corresponding, the relative displacement parameter  is introduced as an indicator for the relative flexibility, which represents the ratio of vertical to transversal displacement for the given load case in 7.3 and hereby allows a comparison of different topology and sandwich designs. In Figure 46the influence of the width of the leaf spring topology structure on the factor  is illustrated for two different sandwich designs for topology A. Both sandwich designs a) and b) are consisting of a core and four symmetric glass fiber epoxy layers with a fiber angle of  and only differ in the core layer. The core layer of sandwich design a) only consists of a thick  carbon fiber epoxy layer, while the core layer of design b) involves an additional  core out of 3D-printer plastic. The connection between the width of the structure and the quality of the obtained component, represented by , is made clearer in Figure 46 for the two sandwich designs a) and b).

Figure 46: Influence of width on absolute displacements and

Figure 47: Influence of width on relative displacement

The topology is now divided into different parts, which are examined and designed separately. As illustrated in Figure 48, the abstracted topology in Chapter 8.3 can be divided into five parts. The first part is representing a long leaf spring bordering the four other sections. Below the influence of the sandwich design on the quality, represented by the relative displacement , is examined step by step for each section.

Figure 48: Topology A divided into 5 sections

8.4.1     Sandwich Design for Section 5

While examining section 5, the sandwich structure designs of the other sections are initially defined as explained below in Figure 49. The influence of the following variables on the quality of the structure is determined in the following:

•     core thickness (Figure 50)

•     fiber angle (Figure 51)

•     Youngs Modulus (Figure 52)

•     number of layers (Figure 53)

Figure 49: initial sandwich designs for examination of section 5

Figure 50: Influence of core thickness on and

As Figure 50 illustrates, the core thickness in section 5 should be reduced to a minimum, which means the sandwich should be designed without any core layer, because the main criterion for identifying a promising topology is the transversal stiffness or rather the horizontal to transversal deformation ratio . The influence of the core thickness in section 5 is a good example for the trade offs coming along with multiobjective optimization design. In this case the increasing braking stiffness ratio  is tolerated due to the improvement of the transversal behavior.

Figure 51: Influence of fiber angle and

Analogously to the given facts illustrated in Figure 50, Figure 51 results in a recommendation to use a  fiber angle for the design of section 5.

Figure 52: Influence of Youngs Modulus on and

Comparing sandwich structures using both, Carbon Fiber Epoxy and Glass Fiber Epoxy layers, it is recommended to use Glass Fiber with a lower Youngs Modulus due to improving transversal qualities illustrated in Figure 52by means of improving .

Figure 53: Influence of number of layers and

Determining the influence of the number of layers on the stiffness ratio, a local optimum can be identified at a number of three layers for the sandwich design in section 5 as pictured above in Figure 53. All four main design parameters for section 5 are summarized in Table 11. It shows either values or upward/ downward tendencies for an optimal design for the characteristic.

Table 11: influence of the design parameters in section 5

The distinctive features as summarized above lead to the following sandwich design in Figure 54.

Figure 54: chosen sandwich design for section 5

8.4.2     Sandwich Design for Sections 2, 4 and 4

The design of sections 2, 3 and 4 is chosen analogously to section 5 in the previous Chapter. The results of the analysis are pictured in Appendix 1.1 in the Figures listed below:

•     core thickness (Figure 68)

•     fiber angle(Figure 69)

•     Youngs Modulus(Figure 70)

•     number of layers (Figure 71)

Table 12: influence of the design parameters in section 2

The influence of the four main variables on the relevant reference value is summarized in Table 12. Finally the chosen designs for sections 2, 3 and 4 as pictured in Figure 55 reasons to assume, that the star-arranged mid sections should be designed for maximum stiffness.

Figure 55: chosen sandwich design for section 2, 3 and 4

8.4.3     Sandwich Design for Section 1

Regarding the illustrated topology in Figure 48, section 1 appears to be the section with the biggest influence on the performance quality of the system. It represents a U-formed leaf spring framing the whole fork. Its design is playing a decisive role in influencing the deformation path and the impact behavior of the whole fork and is also taking the highest loads. For this reason the design of section 1 is carefully and more meticulous chosen at the very last, after examining sections 1, 2, 3 and 4 in the preceding chapters as shown in Figure 55 and Figure 54. In doing so, it is possible to approach the desirable vertical displacement of  for the loads identified in Chapter 7.3. For this matter the following four main design variables are examined in Table 13. The results are derived from the experimental FE analysis of the model, summarized in Appendix 1.2. These examinations lead to the final sandwich designs of all sections, pictured in Figure 56.

Table 13: influence of the design parameters in section 1

Figure 56: final sandwich design

8.4.4     TO of mid sections - 2nd iteration

As proved before, the topology shows the best behavior for maximum stiffness in sections 2, 3 and 4, in the center of the fork. For this purpose an additional TO is carried out for those sections as defined under the influence of the load case identified in Chapter 7.3. The optimization function is set up for minimizing strain and the constrains summarized in Table 14.

Table 14: 2nd iteration TO design constraints

Since the shape and ply stack of the actual leaf spring of the topology is derived in the preceding chapter, the new design domain can be reduced to the one pictured in Figure 57. It follows that the resulting inner topology illustrated in Figure 58seems most appropriate to meet the design for maximum stiffness under the influence of the load case plus the energy absorption characteristics of the leaf spring. In this case the crown region of the fork, which was excluded from the TO design space before, now again is taken into account, because it also is to be designed for maximum stiffness.

Figure 57: 2nd iteration FE-model design space

a)b)

Figure 58: result of 2nd iteration TO a) no load b) deformed

The quality of the result, represented by the absolute displacements ,  and , as well as the relative displacements  and , are summarized below in Table 15.

Table 15: compliant system quality after 2nd iteration TO

8.4.5     CAD modeling

The whole topology forming the connection between the headtube and the main leaf spring is designed for maximum stiffness to support the structure to meet the required dynamic behavior. The two parts of the bicycle fork will be connected to an existing headtube-crone system in order to insure compatibility with common bicycle systems. This leads to the CAD design for the technology demonstrator, optimized for simple prototyping without the cron region, pictured in Figure 59.

Figure 59: CAD model after 2nd iteration

8.5     Shape Optimization

In contrast to the Topology Optimization method the number of elements of the FE mesh is not being changed during the optimization process, as described in Chapter 5.2.5. Shape Optimization marks the last step for the time being before the final CAD modeling phase. During the numerical iteration nodes of the element mesh are moved in order to satisfy the optimization function. The goal is to create a smoother load distribution within the designed structure and do minimize the maximum Mises stress to ensure that the topology will support all loads occurring under normal driving conditions. The shape optimization was performed under planar symmetry and a fixed inner side to avoid contact with the wheel. The objective function was designed to minimize the maximum Mises stress, aiming at an uniform force distribution over the whole leaf spring body. During the Shape Optimization the spring shape was changed as illustrated in Figure 60.

Figure 60: Shape optimization of the leaf spring

8.6     CAD Model Finishing

In this last step the structure is adapted to the results of the shape optimization, which means implementing a variable width of the leaf spring over its length as pictured in Figure 61.

Figure 61: final CAD model

Figure 62 illustrates the weight-optimized structure of the upper fork body core, which is obtained using a simple topology optimization for maximum stiffness, excluding the surfaces from being deleted during the simulation. The side walls of the core body serve as a base material for carbon fiber shear areas.

Figure 62: final core body design in CAD

The final CAD model is examined using the FEM. The resulting Mises stress and vertical displacements are illustrated in Figure 63. The maximum occurring stress is  for the load case introduced in Chapter 7.3, while the maximum displacementis determined at 23,7 mm for a vertical force of.

a)b)

Figure 63: FEM simulation results a) Mises b) vertical displacement

8.7     FE Model Reduction

In order to implement the simulated FE model into an MBS environment, it has to be reduced to rigid links, point masses, springs and dampers, expressed by a global stiffness matrix and eigenvalues, that can be imported in MSC.ADAMS. Since this technique is quite new in computational science, there is no software interface, but an unstrained FE model has to be used for a manual data input to start the reducing simulation. Therefore the Matlab script which can be found in Appendix 1.3 can be used for exporting a FE model into SIMPACK. Slight changes allow a MSC.ADAMS compatible file export. Performing a feasible file export simulation would go beyond the scope of this thesis and is not continued. The global stiffness matrix gained from an FE analysis is pictured below in Table 16.

Table 16: global stiffness matrix [N/mm ; N*mm/°]

8.1           Multi Body Simulation

Further procedure necessitates that the suspension can be evaluated under real driving condition. Multi Body Simulation enables the engineer to test the dynamic behavior of the system and to compare it with traditional spring forks under defined and constant laboratory conditions in the simulation environment. After performing simulation considering all relevant scenarios and conditions, the results give more accurate values for the maximum loads which have to be expected. The suspension system has to be tested and validated using the FEM after implementing the obtained values into the FE model. If the estimated values from the beginning differ significantly from those values, topology optimization is to be carried out again to check if the initial design might also change significantly. Fortunately only the input loads in the existing FE model must be changed.

Figure 64: MBS model of a

For this purpose a cyclist multi body model was modeled at the Department of Mechanism Theory and Dynamics of Machines of the RWTH Aachen using, as pictured in Figure 64. The model was set up in MSC.ADAMS and makes it necessary to export the whole system into Matlab for reasonable computation times. The model covers all the relevant influences, such as gyroscopic forces of the wheels or the pedaling leg. Also the rider is realistically reacting to impacts from the surface by controlling the steering torque and torso leaning, modeled in form of a regulation loop. The results in Figure 65 represent the travelled distance as well as the vertical axle load of the front wheel for a cyclist coasting down on an unprepared path of level 3 suffering short-waved dynamic excitation, using a traditional suspension fork with immersion tubes. In the environment of the introduced model, level 3 means the surface was automatically generated with a wavelength range of , am frequency range of , an unevenness value of  and an unevenness height of  at a length of . [Bre13] These results support the estimated maximal axle load of , derived from the benchmarking as described in Chapter 7.3.

Figure 65: MBS results of the cyclist

In a next step, further analyses considering cornering, braking and accelerating have to be carried out. This phase of the workflow is completed, if all relevant situation are examined for both the traditional fork and the compliant mechanism suspension. To achieve that, the spring damper system in the MBS model simply has to be replaced by the reduced spring from Chapter 8.7. Otherwise these further computations are necessary before prototyping, which would go however beyond the scope of this thesis, because this requires a deeper understanding of the software ADAMS and Matlab.

8.2     Prototyping

Ultimately the designed compliant mechanism fork has to be tried out under real conditions , mounted to a trekking bicycle. Therefore a prototype must be build, using the sandwich design from Chapter 8.4 as pictured below in Figure 66. To allow a manufacturing at a home workshop, the topology will be mounted to a head tube and crown assembly from an existing bicycle. The axle will be attached to a steel bracket that is bolted to an interlaminary screw.

Figure 66: prototype sandwich design

Figure 67: current status of the prototype

The current status of the prototype is pictured above. Manufacturing will be finished within the next week.

9  Conclusion

It is clear, therefore, that the method developed in this thesis on the one hand enables engineers to deal with highly complex multiobjective problem formulations in compliant mechanism design of monolithic suspensions, and on the other hand allows them to quickly gain a general understanding of the nontrivial behavior of the designed system. Knowledge of Finite Element Method and Multi Body Simulation software is assumed. A major advantage of the introduced approach is the expendability of accurate knowledge about the relevant load cases to generate feasible designs. Starting with vaguely estimated forces, more realistic load cases are derived from Multi Body Simulation results later on, if they can be applied to a complete MBS model of the inspected vehicle while travelling. In the following the processed values can be used within further iterations, generating topologies of higher quality at a time, regarding the ability to meet the requirements of suspension systems.

A brief introduction in the state of the art of composite leaf springs, bicycle suspensions and complaint mechanism suspension design in general was given, followed by consolidating the reader’s theoretical understanding of suspension systems, compliant mechanisms and numerical structural design optimization methods. The focus in this work lies on the developed workflow approach, which simplifies the generation of compliant mechanism suspensions using TO and MBS in order to reduce iterations and thus offers a guideline for stringent conceptual drafting of monolithic suspensions. The guideline makes it possible to generate complex topologies feasible for suspension applications.

Using the example of a bicycle suspension fork, it has been demonstrated how iterative software applications can be used to avoid protracted design iterations in monolithic compliant mechanism design by following the workflow approach developed previously. Additionally an example of how to proceed with the resulting topology and the sandwich design of its sections was given. It has been proven that the approach generally offers a guideline to work along with a stringent process in conceptual design. Additionally to the analyses used, further examinations have to be performed, considering for instance fatigue life failure analysis, predictability of FE simulations of composites. Also the dynamic behavior of the system requires eigenvalues of  and damping factors of  should be taken into account. Even though immersing more deeply into the details of composite design has been avoided, the introduced method offers a robust, forgiving and use-oriented guideline to synthesize compliant mechanism suspension systems for concept engineers.

As pointed out in this thesis, the design engineers have to be familiar with the FEM, MBS and CAD software and should have a good understanding of the dynamic situation of the vehicle the suspension system is applied to.

The introduced method is generally applicable to any design problem formulation of a compliant mechanism suspension. Hence it offers design engineers a possibility to validate whether or not the application of a compliant mechanism to a vehicle could be feasible and reasonable. Furthermore the method facilitates the solution to multiobjective and contradictive formulations which derive from resolving the contradictory separation of flexibility and rigidity within suspension systems. The general concept of adding more functions additionally to stability to vehicle frames could lead to broader and more radical concepts. For instance it is conceivably, that suspensions and further functions, such as steering mechanisms through tilting, could be integrated into completely flexible vehicle frames and consequently improve the dynamical performance and weight.

10   Bibliography

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11        Appendix

A1.1    Sandwich Design of Sections 2, 3 and 4

Figure 68: Influence of section 2 core thickness and

>>    maximize core thickness

Figure 69: Influence of section 2 fiber angle on and

>>    0°

Figure 70: Influence of section 2 Youngs Modulus on and

>>    Carbonfiber Epoxy (same thickness! 0,2)

Figure 71: Influence of section 2 number of layers on and

>>    maximize layers

Maximize stiffness for section 2.

Figure 72: Dependence of and on section 3 design

Figure 73: Dependence of and  on section 4 design

A1.2    Sandwich Design of Section 1

Figure 74: Influence of section 1 core thickness on and

Figure 75: Influence of Youngs Modulus on

Figure 76: Influence of section 2 fiber angle on and

Figure 77:    and dz for Carbonfiber-Epoxy layers

Figure 78:  and dz for Glassfiber-Epoxy layers

Figure 79: sandwich design for a CE core

Figure 80: Mises strain for CE core sandwich design

Figure 81: weight of section 1 for CE core design

A1.3    Matlab script for FEM – MBS reduction

%Thorsten.Schrader@rwth-aachen.de

clc

clear

%chosing inputfile

[inputName, inputPath] = uigetfile('*.inp','chose input file','Job-1');

inputName=[inputPath, inputName];

%chosing reportfile

[reportName, reportPath] = uigetfile('*.rpt','chose ABAQUS report file','abaqus');

reportName=[reportPath, reportName];

%chosing outputfile

[outputName, outputPath] = uiputfile('*.inp','place generated input files','Job-1');

outputName=[outputPath, outputName];

outputName=strrep(outputName,'_eigen','');

outputName=strrep(outputName,'_struct','');

outStruct=strrep(outputName, '.inp', '_struct.inp');

outEigen=strrep(outputName, '.inp', '_eigen.inp');

dynRed=input('dynamic reduction (y/n): ','s');

if isempty(dynRed)

dynRed = 'y';

end

if (dynRed=='n')

minEF=input('Minimum frequency of interest: ','s');

if isempty(minEF)

minEF = '0.1';

end

maxEF=input('Maximum frequency of interest: ','s');

if isempty(maxEF)

maxEF = '1000';

end

numberEM=input('Maximum number of eigenvalues to be calculated: ','s');

if isempty(numberEM)

numberEM = '35';

end

else

dynDOF=input('Number dynamic degrees of freedom: ','s');

if isempty(dynDOF)

dynDOF = '40';

end

numberEM=dynDOF;

end;

%format of empty-lines not including a nodal-set definition

leerstelle={[]};

%imports the nodal information from the report file

fprintf('------------------------------\nSTARTING')

%opening the report-file

fid = fopen(reportName);

%reading all numerical data in 4 columns after 19 header-lines

nodeInfo = textscan(fid, '%s %s %s %s','HeaderLines',19);

%closing the report-file

fclose(fid);

nodeStop=strfind(nodeInfo{1,1},'Minimum');

[nodeLines,a]=size(nodeInfo{1,1});

i=1;

while i<=nodeLines

findStop=isequal(leerstelle,nodeStop(i,1));%passing by all empty-lines

if(findStop==1)

nodeNumbers(i,1)=nodeInfo{1,1}(i,1);%saving the node-numbers in seperate vector

nodeX(i,1)=nodeInfo{1,2}(i,1);%saving the nodal x coordinates

nodeY(i,1)=nodeInfo{1,3}(i,1);%saving the nodal y coordinates

nodeZ(i,1)=nodeInfo{1,4}(i,1);%saving the nodal z coordinates

i=i+1;

else

quantityNodes=i-1;%saving the quantity of retained nodes

i=nodeLines+1;

end;

end;

i=1;

%saving nodal informations in an appropriate format (no cell)

nodes=[nodeNumbers nodeX nodeY nodeZ];

fprintf('\n\tsearching node-sets\t\t\t\t00%%')

%searches node-sets and saves their names and number

%temporarily vector 's' includes all rows of the input-file

inp=textread(inputName,'%s', 'delimiter','\n','whitespace',''); lines=size(inp,1);%number of lines in 's'

%finds those lines, in which a nodal-set is defined

set=strfind(inp,'Node Output, nset=');

i=1;

j=0;

k=1;

counter=lines/100;

counter=round(counter);

for i=1:lines

findSets=isequal(leerstelle,set(i,1));%passing by all empty-lines

if(findSets==1)

i=i+1;

else

j=j+1;

%in case of a nodal-set defining line, this line is saved seperatly

setName(j,1)=inp(i,1);setName=strrep(setName,'*Node Output, nset=','');%seperating the set name from the set-defining code of the input file

i=i+1;

end;

if(counter*k==i)%procentual counter

if(k<=9)

fprintf('\b\b\b0%d%%',k)

else;

fprintf('\b\b\b%d%%',k)

end;

k=k+1;

end;

end;

quantitySets=size(setName,1);%saves the quantity of sets

fprintf('\b\b\b\b\tdone\n\tseperating geometry data')

frame=strfind(inp,'** ----------------------------------------------------------------'); %finds the geometry defining frame

i=8;%starts searching after begin frame

for i=8:lines

findFrame=isequal(leerstelle,frame(i,1));%passing by all empty-lines

if(findFrame==1)

i=i+1;

else

delFrom=i;

i=i+1;

end;

end;

fprintf('\t\tdone\n\twriting struct file')

fprintf('\n\t\twriting geometry data \t\t00%%')

fid = fopen(outStruct, 'w');

counter=delFrom/100;

counter=round(counter);

j=1;

for i=1:delFrom

fprintf(fid,'\n');

fprintf(fid, '%s %s', inp{i,:});

if(counter*j==i)%procentual counter

if(j<=9)

fprintf('\b\b\b0%d%%',j)

else;

fprintf('\b\b\b%d%%',j)

end;

j=j+1;

end;

end

fprintf('\b\b\b\b\tdone\n\twriting simulation data')

fprintf(fid,'\n** \n**SUBSTRUCTURE PROPERTY, ELSET=SUBSTRUCTURE \n** Calculation of dynamic modes---------------------- \n*step \n*frequency,eigen=lanc \n');

if (dynRed=='n')

fprintf(fid,'%s',numberEM);

fprintf(fid,', %s',minEF);

fprintf(fid,', %s, , ',maxEF);

else

fprintf(fid,'%s',dynDOF);

end;

fprintf(fid,', \n*boundary');

for i=1:quantitySets

fprintf(fid,'\n');

fprintf(fid,'%s,1,6 %s',setName{i,:});

end;

fprintf(fid,'\n*output,hist,freq=0 \n*end step \n*FILE FORMAT, ASCII \n*STEP \n*SUBSTRUCTURE GENERATE,TYPE=Z001,RECOVERY MATRIX=NO,MASS MATRIX=YES,OVERWRITE,LIBRARY=SIMP \n*SUBSTRUCTURE MATRIX OUTPUT, MASS=YES, STIFFNESS=YES \n*RETAINED NODAL DOFS,SORTED=NO');

for i=1:quantitySets

fprintf(fid,'\n');

fprintf(fid,'%s,1,6 %s',setName{i,:});

end;

fprintf(fid,'\n**RETAINED EIGENMODES, GENERATE \n*SELECT EIGENMODES, GENERATE \n1, ');

fprintf(fid,'%s',numberEM);

fprintf(fid,', 1 \n** \n*END STEP');

fclose(fid);

fprintf('\t\t\tdone\n\twriting eigen file')

%writes the 'eigen'-file

fid = fopen(outEigen, 'w');%creates the output file to write into

fprintf(fid, '** To perform Eigenvalue analysis \n*NODE \n');%writing ABAQUS input-code

for i=1:quantityNodes

fprintf(fid, '\t\t %s, ', nodeNumbers{i,1}');%including the retaining nodes

fprintf(fid, '\t\t %s, ', nodeX{i,1}');%including the retaining nodes

fprintf(fid, '\t\t %s, ', nodeY{i,1}');%including the retaining nodes

fprintf(fid, '\t\t %s \n', nodeZ{i,1}');%including the retaining nodes

end;

%writing ABAQUS input-code

fprintf(fid, '*ELEMENT,, ELSET=SUBSTRUCTURE, FILE=SIMP \n1000');

fprintf(fid, ', %s', nodeNumbers{:,1});%including the retaining nodes to be abstracted into a substructure

fprintf(fid, '\n** \n*SUBSTRUCTURE PROPERTY,ELSET=SUBSTRUCTURE \n** Set boundary conditions if any... \n** \n*FILE FORMAT,ASCII \n** modal analysis ---------------------------------------------- \n*STEP \n*FREQUENCY,EIGENSOLVER=LANCZOS,NORMALIZATION=MASS \n');

if (dynRed=='n')

fprintf(fid,'%s',numberEM);

fprintf(fid,', %s',minEF);

fprintf(fid,', %s, , ',maxEF);

else

fprintf(fid,'%s',dynDOF);

end;

fprintf(fid,',\n*NODE FILE,GLOBAL=YES\nU \n*EL FILE \n*END STEP');

fclose(fid);

fprintf('\t\t\t\tdone\nDONE\n------------------------------\n')