Project 3.13: Systems Approach to Wheel and Pad Metamaterial Design Including Robustness Issues PI: Fadel (Clemson)

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1 PI: Fadel (Clemson) Project started: 2017 Estimated end: 2018 Resources / Funded effort: 2017 PI 1 SM, FAC 1 SM, 1 GSRA key: PI Principal Investigator (faculty unless otherwise indicated) co-pi co-principal Investigator (faculty unless otherwise indicated) FAC faculty quad member RS research scientist PD post-doctoral student GSRA graduate student research assistant RA/temp temporary student assistant (part time) SM summer month AY 9 months basis CY calendar year 12 month basis Eqpt equipment (> $5,000 requires executive committee approval) note: research supplies and travel are not listed

2 Project 3.13: ARC NEW PROJECT PROPOSAL Project 3.13: Systems Approach to Wheel and Pad Metamaterial Design Including Robustness Issues Quad Members: Name Georges Fadel (PI) Gang Li Samuel Franklin Matthew Castanier David Ostberg Bill Bradford Christopher Cardine Position, Affiliation Professor, Clemson University Professor, Clemson University Graduate Student, Clemson University Research Mechanical Engineer, US Army TARDEC Engineer, US Army TARDEC R&D Scientist, US Army TARDEC General Dynamics Land Systems Contact Project Duration: This research project will start in January We estimate the tasks in this research project should be accomplished in two years. Motivation and Background: For the last four years, Clemson has worked on designing the metamaterial pad for the Tank track and is currently testing it. Figure 1. shows the current design and its predicted compressive deformation behavior compared to that of the original elastomeric pad. Our initial study enabled us to design a titanium pad that replaces the elastomeric pad, and that deforms similarly to the elastomeric pad under a static load. In fact, Fig. 2 from [3] shows that the whole system, wheel and pad should be considered if this problem is to be holistically addressed. Fig, 1. Ti pad designed by Clemson Fig. 2, Tank track temperatures under exercise (taken from [3]). 99

3 Fig. 2 shows that the elastomer on the wheel is also red hot, and may eventually break. Both the pads and the road wheel should therefore be considered as part of an overall system to better understand the problem and to optimize the design. Coupling both components would allow us to (1) eliminate the heatgenerating hysteretic losses of the elastomeric layer on the wheel, (2) coordinate the deformation between the two components of the suspension, and most importantly, (3) we should be able to design the wheel significantly lighter than the current wheel, but with the same performance level otherwise. Introducing the aspect of robustness will enable the consideration of aleatory aspects and design a system that is less sensitive to variations in these aspects. Why is this important? In the previous study, we considered the total weight of the tank distributed in the road wheels and on the pads in contact with those wheels. The pads were therefore designed to carry their respective load on their whole surface. The force applied on the pad was the same as the one used in the Army calculations. In fact, as the wheels deform when in contact with the pads, the contact surface between both components dictates the magnitude of the pressure. For the flat titanium pad, the deformation is minimal, and thus the results obtained to date should be valid, but the wheel deforms, and the surface of contact needs to be computed then transmitted to the pad design. This coupling is what we plan to compute and use in the metamaterial design of both the wheel and the pad. This coupling also opens up a new design space and allows us to explore tuning the static and dynamic characteristics of the system such as the weight, vibration and harshness performances. Research Objectives: Building upon the Unit Cell Synthesis method we have developed for the design and topology optimization of the metamaterial track pad, the main research objective of this project is to develop a methodology that can be used to design a system of two coupled meta-material components that meet some specific elastic requirements. The method will be demonstrated on the design of the tank track pads and road wheel system, as illustrated in Fig. 3. Now that the pad has been designed, the road wheel has to be considered. This road wheel does not have the manufacturing constraints that the pad had, since the whole wheel outside a hub can be considered as the metamaterial. In this case, a casting of the wheel would be perfectly acceptable, and additive manufacturing will not have to be considered. Approach: Designing and optimizing of the tank track pad and road wheel system containing two coupled metamaterial components (pad and wheel) is challenging due to the following reasons: (1) the targeted system level mechanical behavior is the result of the two interactive metamaterial components. While the coupling offers more flexibility of the metamaterial design, it also largely increases the degrees of freedom (DOF) of the optimization problem; (2) our previous study has assumed that the wheel is pushing the whole pad, and we did not consider any other deformation than the one due to compression. With the wheel, however, compression and shear will be present. This in itself is a challenging problem since we have not seen anyone attempting to design a material that matches two loading conditions simultaneously; (3) the dynamic interactions between the components, such as vibrations, impact and damping ratio, are important in the performance of the pad-wheel system. They must be considered in the optimization process; (4) the uncertainty in the manufacturing processes, and the variation of various dimension should be addressed in the optimization process for robust design of the system. Thermal issues should be also addressed to show that the modified system is more able to reject heat than the current one. 100

4 Figure 3. Artist concept of coupled wheel-pad system To address the challenges and realize the objectives of this research, we propose the following activities: Task 1: Generalizing metamaterial optimization methods from component to system level. In our previous work, we have developed a Unit Cell Synthesis method for the design of the pad with targeted nonlinear compressive deformation response. By using a nonlinear spring network analogy, the method defines and connects a set of Elemental Functional Geometries (EFGs) to form a unit cell that possesses the target deformation characteristics. Multi-objective optimization is then employed to optimize the initial design of the EFG connections to match the target stress-strain curve. For the pad-wheel system, the overall system problem will be decomposed into the wheel and the pad sub-problems, each subsystem is constructed using unit cells composed of EFG networks, and the two optimizations of metamaterials will be coordinated using the augmented Lagrangian coordination method. The coupling variable will be the contact surface between them. The load on the track depends on the deformation of the wheel that dictates the patch size, and therefore the pressure on the pad, whereas the pad resistance will push back on the wheel to put the system in equilibrium. Task 2: Generalizing metamaterial optimization methods to include mixed mode deformations. Due to the applied torque on the wheel and the rolling resistance, significant shear exists in the wheel along with the compression. The modeling of the wheel metamaterial will have to consider the coupled shear and compression effects, and the material will have to be designed to match the elastomer in both directions. Therefore, the metamaterial optimization at the wheel will attempt to generate a topology that is optimal for a combined loading case, whereas the pad has been designed only for compression. For the design of metamaterial wheel with combined loadings, the Unit Fig 4. Unit cell having a 2-D EFG network. Cell Synthesis method will be extended to include shear deformation. Specifically, instead of using 1-D nonlinear spring network analogy to form the unit cells, the geometric nonlinear deformation of the EFGs will be connected to form a 2-D network, as shown in the Fig. 4. Such 2-D EFG network based unit cells will be categorized into 0, 1, 2 n order systems. The axial and shear deformation characteristics of the predefined systems will be pre-calculated and stored in a database. For a given target deformation response, the unit cell designs that qualitatively match the target response will be selected as candidate designs. Then the multi-objective optimization will be carried out to optimize the dimensions of the candidate designs. Note that, in the Unit Cell Synthesis method, multiple solutions of the problem are possible. The coupling between the two modes of the deformation will enable us to get the more accurate displacement of both sub-components, and will more accurately model the system behavior. 101

5 Task 3: Generalizing metamaterial optimization methods from targeting static stress-strain behavior to dynamic responses. In the track pad-wheel system, the dynamic response of the wheel, such as the vibrational frequencies, maximum dynamic displacement, contact stress during impact loading, etc., is important for the operation of the tank. One of the advantages of the Unit Cell Synthesis method is that multiple performance measures can be computed for any EFGs connections in the unit cell. The response characteristics can be recorded and stored for future selections to match the target response. In this work, we will generalize the Unit Cell Synthesis method to include some of the dynamic responses as performance measures and as a part of the objective function of the size optimization. Task 4: Enhancing the robustness of metamaterial system design and optimization. The issue of robustness will consider the uncertainty in the manufacturing processes, and the variation of various dimension. We will combine our Unit Cell Synthesis method with the reliability based design optimization (RBDO) approach to address this issue. In this regard, we will continue our collaboration with Prof. K. K. Choi at University of Iowa and attempt to utilize their RBDO tools in the optimization process for the track pad-wheel system design. Task 5: Baselining unit cell synthesis approach versus classic topology optimization methods - performance of final design. Currently, topology optimization is the most popular method for designing new meta-materials. Topology optimization has been used along with an inverse homogenization approach to target prescribed material properties while minimizing the cost (e.g. volume). However, the inverse homogenization problem involving geometric nonlinearity must consider multiple prescribed stiffness tensors that are dependent on the loading condition. These stiffness tensors represent the tangents to the nonlinear target response. Due to the added complexity of multiple tangent prescribed stiffness tensors, implementation of the nonlinear inverse homogenization problem is nontrivial. Although designing metamaterials with prescribed nonlinear response (Poisson s ratio) using topology optimization has been reported recently, the application of the technique for prescribed nonlinear mixed mode deformation remains unexplored and there is no computational tool available for such optimization problems. In this work, parallel to the Unit Cell Synthesis method, we plan to implement the topology optimization approach for the design of the pad-wheel system. We propose to obtain optimized the designs of the system by using both approaches and the performance of the final designs will be evaluated and compared. The evaluation and comparison would not only guarantee the best design of the padwheel system but also serve as a case study baselining the unit cell synthesis approach versus topology optimization approach. Note that we envision that the wheel will consist of three parts, the hub, solid steel, the transition area which could be some honeycomb structure, and the elastomer equivalent metamaterial which is in contact with the metamaterial pad. The coupling of the two wheel and pad components into one system will be accomplished by mathematically connecting the two optimizations and analyses using the augmented Lagrangian methods described in our most recent paper by Xu et al. which is listed in our recent publications. One more note: the other aspect that can be managed much better with the proposed system is the heat transfer problem. Contrary to the insulating rubber, the metamaterial is a good thermal conductor, and we therefore will be able to prove through simulation that the temperature rise on the wheel will be significantly lower than with the current system. This part of the work will occur once the wheel pad system has been designed and optimized. 102

6 Proposed Timeline: Q1 Q2 Q3 Q4 Y2 Year One Research Tasks and Deliverables Developing wheel geometry and attempting topology optimization of couple deformations. Continuing wheel topology optimization, and deciding on the thickness of the metamaterial layer to accomplish the deformation while reducing material and weight. Develop approach to couple wheel and pad optimizations. Introduce thermal modeling Start the coupled wheel pad optimization in tandem. Long-term goals Address the system level approach of the design of metamaterials considering manufacturing constraints and robustness issues. ARC/TARDEC/Industry Benefits: Project needs to demonstrate the potential in adding value in at least one of the following areas: 1) Interactions with and benefits to other ongoing ARC research 2) Interactions with and benefits to ongoing TARDEC work 3) Interactions with and benefits to ongoing work at Industry partner TARDEC has asked us to consider the system level approach to the metamaterial design problem. This project builds on a previous ARC project, and should entice GD to be more active and supportive of the work. Leveraged Funding: No current leveraging exists. Qualifications: Drs. Fadel and Li have proven their ability to address the tank track pad problem, and plan to build on this success. Dr. Fadel has also worked on the Michelin Tweel project which was supported by a NIST ATP and Michelin, and consisted in designing a metamaterial that matched rubber shear stiffness in the nonpneumatic tire of Michelin. The objective was to reduce or eliminate the hysteretic losses due to the rubber deformation, and therefore to reduce rolling resistance in tires. Publications from Prior Work closely related to the proposed project: Satterfield, Z., Kulkarni, N., Coutris, N., Li, G., Fadel, G.M., Castanier, M, Unit cell synthesis method to design meta-materials with targeted nonlinear deformation response paper submitted to the J. Mech. Des, 2016 Czech C., Guarneri P., Thyagaraja N. and Fadel, G,M. Systematic Design Optimization of the Metamaterial Shear Beam of a Nonpneumatic Wheel for Low Rolling Resistance, This paper (listed below) was selected as a feature article in the April 2015 edition of the J. Mech. Des. Companion Czech C., Guarneri P., Thyagaraja N. and Fadel, G,M. Systematic Design Optimization of the Metamaterial Shear Beam of a Nonpneumatic Wheel for Low Rolling Resistance, J. Mech. Des. 137(4), (2015) (9 pages); Paper No: MD ; doi: / Czech, C., Guarneri, P., Wiecek, M.M., and Fadel, G.M., Optimality Conditions for Topology Optimization Problems Targeting Meta-material Properties, Accepted for publication Structural and Multidisciplinary Optimization Journal, April 2013 Xu, M., Fadel, G.M., and Wiecek, M., Investigation of Variants of Augmented Lagrangian Coordination and Extension of the Dual Residual Theory, paper submitted to the J.Mech.Des, May