Optimal Design of a CompliantRearMountain Bike Suspension

Semester Project Final Report

ME 558 Discrete Optimization

Submitted 11/13/2018

OptiCycle

James T. Allison

michael S. Cherry

Zachary A. Kreiner

Abstract

The objective of this project was to determine a novel topology for a compliant rear mountain bike suspension by employing discrete and continuous design optimization methods. The mathematical model we have developed accounts for horizontal and vertical loading on the wheel hub, the path the rear wheel hub travels while in motion, the total horizontal and vertical displacement of the hub, and the volume of the suspension. For simplicity, we began with a nine-node mesh representing the possible design space, but then increased the mesh resolution, demonstrating the scalability of our parametric model. The final model is a sixteen-node, undirected graph where the elements connecting the nodes indicate the possible orientation of beams that compose the rear suspension. We have designed and built a custom Matlab code to analyze this space and minimize a multi-objective function consisting of the path error with respect to a circular path about the bottom bracket, the suspension volume, and the horizontal displacement. Using this optimization program, a Pareto surface was then generated to illustrate the tradeoffs between the multiple objectives. With this code, we have been successful in determining an optimal topology that is lightweight, very stiff in the horizontal direction, and closely follows the desired path.

Table of Contents

Problem Statement

Design Model

Mathematical Model

Definition of Design Variables

Definition of Constraints

Definition of the Objective Functions

Optimization Method

Optimization Model Description

Optimization Results and Discussion of Results

Conclusions

References

Appendix A: Optimization Code Structure

Appendix B: The Matlab Optimization Code

Problem Statement

In today’s mountain bike market, the rigid rear triangle seen in bicycles for generations has been widely replaced by a rear suspension to provide a smoother ride while trail-riding and increased control on rough terrain. This trend towards high-performance cycles has substantially increased the cost of the mountain bikesdue to the assembly costs and part manufacture inherent with rigid-link mechanisms. Much work has been done in developing advanced models for full suspension mountain bikes [1,2,3], most of which focus on rigid-link mechanisms. Team OptiCycleproposed that a novel compliant mechanism could be designed to replace theserigid-link rear suspensions in a simple and cost-effective manner with a compliant suspension. The rear triangle of a mountain bike (See Figure 1, below) has many product requirements that work against one another, making it an excellent choice for an optimization problem that implements compliant mechanisms. These requirements are as follows:

  • The compliant suspension should attach rigidly to a given seat tube geometry.
  • The suspension should be lightweight.
  • The suspension should be reliable, i.e.- exhibits a high fatigue life, high corrosion resistance, and high strength.
  • The hub should travel a prescribed path while under both horizontal and vertical forces, as well as torsion.
  • The dynamic response of the rear suspension should enhance mountain bike control and comfort.
  • Finally, the ergonomics and aesthetics of the suspension should be considered.

Figure 1: A diagram indicating which section of the mountain bike will be modeled for the creation of a novel compliant mechanism

These requirements provide an ideal topology optimization problem due to the conflicting nature many of them have with one another. Mountain bikes are used under extremely rough conditions, making strength a key component in any design, yet the carrying weight of the bike should be as low as possible, making heavy and overly strong components infeasible. Also, the suspension has to be stiff enough to restrain the forces induced during normal use, while at the same time it must be compliant enough to provide the correct amount of deflection under a prescribed force. These inverse relationships increase the difficulty of building a suspension, and necessitate the aid of computer optimization to find an acceptable design.

While each of the considerations is important in the design of a compliant rear mountain bike suspension, many of them were beyond the scope of the project. In this light several simplifying assumptions were made. For this initial problem, dynamic effects, fatigue, buckling, and stress have been ignored. These are acceptable assumptions to make, because they are intimately related to the stresses inside the geometry, and stress is typically ignored during the topology optimization stage for compliant mechanisms[4]. To further simplify the problem, we assumed the problem was two-dimensional, with a uniform depth. This means that torsion and lateral rigidity (stiffness out-of-plane under torsion about the seat tube) will also be ignored. In the beginning, aesthetics and ergonomics were also ignored, but as possible topologies were proposed later, they would again be considered based on the opinion of the team members. One final simplification is to constrain the material to 4130 steel, removing a degree of freedom from the model.

With these considerations removed, the problem was formulated with three objective functions and one constraint. First, the path accuracy would be measured and minimized against a circular path using a sum squared error approach. Second, the volume would be minimized, as well as the deflection in the horizontal direction under a substantial horizontal load, as is the case for braking. The single constraint was that any considered topology should deflect 5cm under a maximum force of 2400N, the force of an average rider under 3gs where g is the acceleration of gravity (9.8 m/s2). This would ensure that the suspension would have the correct deflection during trail-riding. More deflection was not considered because it would send the system into the non-linear region, increasing the computational time by as much as 2000 times.

Design Model

With this formulation, the design space was set up as a grid of nodes connected by elements indicating the possible paths the suspension could travel from the rear hub to the seat tube, and the initial nine-node, eighteen-elementspace used to replace the rear triangle can be seen below in Figure 2. Each element in the space can be turned on or off, changing the design. Once our design method worked on this simple mesh, we planned to refine the mesh until the design model approximated a continuous design space, allowing for complex designs such as curves and tapered beams. Discrete optimization is ideal for such a formulation, because it allows for the simple addition or removal of these elements. The problem would also include continuous optimization in the form of the thickness of the elements.

Figure 2: The initial design space, a nine-node, eighteen-element system

One challenge presented by this design model was system connectivity. For a design to be valid, the rear wheel hub must be physically attached to the seat tube at least at one point. A disconnected, invalid design can be seen in Figure 3, below. Such disconnected designs were dealt with in two different fashions which will be explained later.

Figure 3: A disconnected, invalid design

Mathematical Model

The model, design objectives, constraints, and variables can now be presented formally in negative null form:

Definition of Design Variables

The variableis a binary vector. When the element based design model is used, each element of represents the existence ofthe corresponding element within a design. (0 → doesn’t exist, 1 → does exist). This allows for direct design modification by adding or removing elements. When the path based method is used, each element ofinstead describes the existence of a singly connected path. This allows for indirect design modification by adding or removing paths.One advantage of the path based method is it guarantees connectivity of the design, whereas the element method does not.

Beams with equal depth and thickness were used to describe all the elements in each design. The continuous scalar variable indicates this value. (All beams had the same depth and thickness.) In future work, a vectorcould be used instead to allow each of the elements to have a different thickness.

The weighting parameter wi is necessary for the optimization model, and is not a true design variable. It will be described in detail in the definition of the objective function.

Definition of Constraints

The only explicit constraint for our optimization model is an equality constraint that requires the maximum vertical deflection attained by each design (δmax) when subjected to a prescribed force is equal to the target deflection (δtarget). If the deflection is too low, the design won’t have the correct stiffness, and if the deflection is too high, the design will travel into the non-linear range, invalidating our computer model. The FEA code designed by our team varies the thickness of all the beams until a value is found that meets this equality constraint for a particular design topology.Because only one thickness value will meet the constraint for a particular design, discrete choices are imposed. In addition, because of this equality constraint, t may be viewed as a dependent variable, rather than an independent design variable. The only true independent design variable is.

Other discrete constraints are imposed implicitly. corresponds to the existence of elements in the predefined design space. An element outside of the design space cannot be selected. This implicit constraint imposes discrete design variable options.

Definition of the Objective Functions

We have set up this design problem with three objectives to be minimized; the path error, the volume, and the horizontal deflection. These objectives ensure the optimal design will have the necessary specifications for a rear frame and suspension with respect to the scope of this project.

It was desired that the hub of the rear wheel should travel in a circular path about the bottom bracket of the mountain bike. This circular path ensures no force feedback through the chain when the suspension is active, and reduces suspension bobbing resulting from pedaling forces. Each design was placed under five load steps, and the position of each point was evaluated against a circular path. The sum squared error of these evaluations was used as the first objective function to be minimized.

For optimal suspension performance, the massof the design should be as small as possible. Since mass is proportional to system volume, volume was used as the second objective function to be minimized.The length of all the existing elements was found, and multiplied by the square of the uniform thickness to calculate the system volume.

The horizontal deflection is the longitudinal movement of the rear wheel hub (parallel or antiparallel to the direction of motion) that occurs when longitudinal forces are applied to the rear wheel (braking or acceleration). The horizontal deflection was used as the third objective function to be minimized. It should be small because as the hub moves horizontally, the length of the wheelbase changes, and results in poor handling.

Since three objective functions are used, an approach must be used to convert the three objective functions to a single value, since a vector cannot be minimized. Two popular methods are to either create a composite objective function by summing the weighted values of the individual objective functions, or by choosing an objective function, and converting the remaining objective functions to constraints. We chose to use the former method. wi is a weighting parameter used to vary the importance of each of the objective function values. wi is continuous, and is determined using a normalization method to ensure each set of wi values results in a unique optimal design. A collection of these different optimal designs found using different weighting factors can be used to generate a Pareto surface that will be described later.

Optimization Method

The nature of the problem undertaken requires the use of a heuristic algorithm. There is no continuous counterpart to the topology optimization problem. In the approach used in this project, the existence of each element in the node matrix is easily represented as a single binary number. The string of all the binary numbers then completely describes the topology of the structure, or design. This type of binary representation lends itself well to using a genetic algorithm (GA) for optimization. The other alternative we considered was simulated annealing (SA). In general it seems that SA requires a much larger number of calls to the analysis function than required by a GA. The analysis function consists primarily of assembling the stiffness matrix for the specified topology, and applying various force loads, returning the resulting displacements. This function required approximately 0.4 seconds to execute, and thus was a limiting factor in our choice of optimization methods and parameters.

As mentioned previously, a unique challenge faced in this topology optimization was maintaining connectivity between the rear hub where the wheel is mounted, and the seat tube on the rigid frame. Two techniques were addressed to handle this problem, the path and the element methods for determining element existence. The path method guarantees connectivity by considering each possible connected path from the rear hub node to the nodes of the seat tube as a design variable. The element method considered each possible element connecting two nodes to be the design variables, and therefore does not ensure connectivity.

In order to use a path method, each possible path from the rear hub to the seat tube needed to be enumerated. This was accomplished by creating the adjacency matrix for a given mesh, and running a modified one-to-some depth-first search algorithm to find all possible paths and store them in a matrix. (See Figure 4)

Figure 4: The path method builds the suspension from combinations of paths

For a 3x3 node mesh, there were 113 possible paths, and thus the optimization required 113 binary design variables, representing the existence of each path in the specified design. Before running the analysis function, an additional step was required in which the set of existing paths was decoded into the existing elements by taking the union of all the elements in the existing paths for a single design. Figure 5, below shows one optimized 3x3 design space.

Figure 5: The optimal 3x3 topology found by the element method

A problem faced with this path method was densification of the designs in the population. Any more than four or five existing paths resulted in a truss structure with very few if any elements missing from the mesh. This problem was furthered by mutation and crossover, because changing just one design variable could drastically alter the topology. Such drastic changes made convergence on a global optimum difficult, if not impossible. Another problem with the path method was its lack of scalability. The resolution of the mesh must be increased if a viable compliant rear suspension is to be found, but to increase the mesh to 4x4, the binary existence variables increase to 20,596. This is far too many to be optimized with a genetic algorithm, so we had to find a different method.

To solve these problems, an element method for determining element existence was devised. This method solved the problem of densification because each design variable only indicated one element in the space, rather than an entire path in the space. This way, mutations and crossovers would change the topologies less drastically, enabling convergence on a global optimum. The optimal 3x3 design found with the element method can be seen in Figure 6, below.

Figure 6: The optimal 3x3 mesh found by the element method

The issue of scalability was also addressed by the element method. With each binary existence variable indicating only one element, the number of variables for a 3x3 mesh was reduced from 113 to 18, and for the 4x4 case it dropped to only 39 from 20,596. The element method, however, does not ensure connectivity like the path method. Because of this a check for connectivity had to be added. This is a simple check that will be described later. Despite the necessity of this check, and due to its scalability and ability to converge, we chose to continue the suspension project with the element method.

Once the decision had been made to use a genetic algorithm along with finite element analysis, we had to create an optimization routine in which we could implement them. Figure 7 illustrates the 4 methods we originally considered for our routine.

Figure 7: Anillustration of the four solution methods considered

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