Current Projects

Below, you will find a sample listing of some of the research projects taking place within the Mechanical Engineering department. Use the search box or advanced filtering options to search our research projects by keyword or by investigator. You may also learn more about our research thrusts and the projects related to each area:

Research Thrusts


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Modeling and Control Development for Electric Vehicle and Smart Grid Integration

Investigators
Principal Investigator: Bo Chen
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Oculus - ASR Nanosatellite Flight Integration

Investigators
Principal Investigator: Lyon King
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Collaborative Research: On Making Wave Energy an Economical and Reliable Power Source for Ocean Measurement Applications

Investigators
Principal Investigator: Umesh Korde
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Hydrodynamic Control Using X-Band Radar for Wave Energy Converter Technology

Investigators
Principal Investigator: Umesh Korde
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

I/UCRC: Novel High Voltage/Temperature Materials and Structures

Investigators
Principal Investigator: Gregory Odegard
Co-PI: Julia King
Co-PI: Paul Sanders
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Advanced Controls in Wave Energy Conversion

Investigators
Principal Investigator: Ossama Abdelkhalik
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Natural Gas Research with Argonne National Laboratory

Investigators
Principal Investigator: Scott Miers
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Increasing Ship Power System Capability through Exergy Control

Investigators
Principal Investigator: Gordon Parker
Co-PI: Rush Robinett
Co-PI: Eddy Trinklein
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Collaborative Research: On Making Wave Energy an Economical and Reliable Power Source for Ocean Measurement Applications

Investigators
Principal Investigator: Ossama Abdelkhalik
Co-PI: Rush Robinett
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Ignition System Characterization for Chrysler

Investigators
Principal Investigator: Jeffrey Naber
Co-PI: Jaclyn Johnson
Co-PI: Seong-Young Lee
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics
Project Description and Research Objectives:
Michigan Technological University (MTU) will investigate and characterize Fiat Chrysler Automobiles (FCA) supplied ignition systems to provide ignition characteristics and data for FCA model validation under various ambient conditions. Tests will be conducted in Michigan Tech’s optically accessible Combustion Vessel (CV) research facility. The ambient conditions that will be varied include Air-Fuel Ratio (AFR), charge velocity, and pressure at ignition. Limited testing will also occur to evaluate the impacts of spark plug orientation. Particle Image Velocimetry (PIV) will be conducted in the CV, without the presence of an igniter, to quantify charge velocity. A total of three ignition systems will be tested. Ignition system calibration settings will be defined by FCA. All testing will be conducted with propane fuel. Results will include high speed Schlieren imaging to quantify flame kernel development and flame propagation. Results will also include ignition system measurement of primary and secondary current and voltage ignition System Characterization for Chrysler

Awarded Amount: $84,695

High Brake Mean Effective Pressure (BMEP) and High Efficiency Micro-Pilot Ignition Natural Gas Engine

Investigators
Principal Investigator: Jeffrey Naber
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics
OBJECTIVES:

The objective of this project is to develop the combustion system for a low-cost, low diesel contribution, high brake mean effective pressure (BMEP), high-efficiency premixed charge medium/heavy duty (MHD) natural gas engine and demonstrate the technology on an engine with peak thermal efficiency of up to 44%, diesel pilot contribution of 1-5%, and BMEP up to 25 bar. Emissions will be compliant with current Environmental Protection Agency (EPA) standards for heavy-duty (HD) on-road engines by using a three-way catalyst.

SCOPE OF WORK:
This project will evaluate and develop solutions to the barriers to micro-pilot combustion in a stoichiometric natural gas engine. A combination of combustion vessel (CV) testing and computational fluid dynamics (CFD) simulation will evaluate fundamental limitations and develop solutions. The results will then be applied to a multi-cylinder engine test bed, where the combustion system will be developed and emissions and efficiency demonstrated.
The project will be conducted in 3 budget periods:
Budget Period 1: Fundamental study of micro-pilot NG ignition
In BP 1, prior art on micro-pilot natural gas (NG) ignition will be investigated, CV will be set up and used to conduct some preliminary study on micro-pilot NG ignition, combustions models will be validated using existing data, and engine baseline will be established.
 Budget Period 2: Development of micro-pilot NG combustion
In BP2, more CV testing will be conducted to further develop the micro-pilot NG ignition concept.
Engine testing will then be carried out using the promising design and operating conditions determined by the CV and CFD simulation.
Budget Period 3: Optimization of micro-pilot NG engine
In BP3, micro-pilot NG engine concept will be further optimized on the engine with the help of both CV testing and CFD simulation. Based on that, design specifications will be provided and the readiness of the technology for commercialization will be assessed.

Awarded Amount: $1,229,000

Auris: A CubeSat to Characterize and Locate Geostationary Communication Emitters

Investigators
Principal Investigator: Lyon King
Co-PI: Ossama Abdelkhalik
Co-PI: Michael Roggemann
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Autonomous Microgrids: Theory, Control, Flexibility and Scalability

Investigators
Principal Investigator: Wayne Weaver
Co-PI: Nina Mahmoudian
Co-PI: Rush Robinett
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Stratus: A CubeSat to Measure Cloud Structure and Winds

Investigators
Principal Investigator: Lyon King
Co-PI: Ossama Abdelkhalik
Co-PI: Michael Roggemann
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Stratus, a NASA CubeSat, and the Utilization of Effective Project Management to Enhance Student Learning

Investigators
Principal Investigator: L. King
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics
Abstract
In order to maximize the student learning experience through a project that consists of the design, test, and fabrication of a cubesat, Sam Baxendale proposes conducting research into effective Project Management skills for student satellite teams. Sam Baxendale will serve as Project Manager of Stratus, a NASA cubesat proposed to image cloud movement from geostationary orbits in order to optimize solar power generation applications. Managing a team of 60 undergraduate Michigan Technological University Students, Sam Baxendale will work with Faculty Advisor Dr. Lyon Brad King to promote an environment in which students are presented the opportunity to gain hands-on experience through the development of a spacecraft that will be ultimately launched and utilized to serve the strategic interests of NASA.

Awarded Amount: $2,500

Senior Design: Flywheel Balance Measurement System

Investigators
Principal Investigator: William Endres
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics
Project Goal
Create a flywheel balance measurement process that yields improved performance versus currently available methods and equipment.
 
Background
The current best industry practice for measuring the imbalance of flywheels produces results that are inconsistent and has insufficient sensitivity. Mercury has not been able to identify equipment that can demonstrate statistically acceptable results for Repeatability and Reproducibility (R&R). It appears that the flywheel balancing process is not completely understood by suppliers currently providing balance measurement equipment.
Project Scope
This project will focus on identifying a methodology to measure the imbalance of a single mass marine flywheel within a set weight and diameter range. The design team on this project will initially research past and current methods of measuring imbalance initially starting with focusing on other rotating assemblies outside the current flywheel methods. Based on research results, the team will devise a concept for the measurement process, construct a prototype, and use the prototype to produce data demonstrating validity of the concept.
 The existing methods for flywheel balance measurement are evolutionary and very similar to one another. These methods have demonstrated low repeatability and have low accuracy relative to some existing part tolerances. A new method will be developed with an alternative technology and not focusing on current practices.
Project Objectives
• Design concept for a flywheel balance measurement system
• Prototype unit based on the design concept
• Data set demonstrating concept validation

Awarded Amount: $25,650

On Integrating New Capability into Coastal Energy Conversion Systems

Investigators
Principal Investigator: Ossama Abdelkhalik
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics
Overview:
MTU will analyze and simulate the power capture from arrays of wave energy converters (WECs) with and without the presence of an object. Nonlinear WECs will be analyzed and exploited for more energy capture. For object detection, MTU will develop an estimator. In addition to having a model that detects the presence of an object, the estimator will use that model and account for uncertainties that we have in the model and also measurement errors; in any case we need to know statistical characteristics about these uncertainties and errors. MTU will participate in the WEC array overall design, analysis, modeling and simulations; control design for Design 2, nonlinear modeling and control, and topology optimization.

Awarded Amount: $405,139

Numerical Simulation Study of Post Collision Angles for Multiple Impinging Jet Injectors

Investigators
Principal Investigator: Seong-Young Lee
Co-PI: Jeffrey Naber
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics
Objective:
The objective of the project is to perform simulations on multi-hole impinging jet injectors under different injection pressures and ambient conditions as well as number of hole to explore these effects on post collision angle. Also, the spray behavior and spray characteristics will be studied using
Michigan Tech established spray models. The primary aim of this simulation is to study the trend of post collision angles with changes in the impinging collision angle with various number of injector holes. The project includes two-step simulations, i.e., the first stage and second stage. The first stage covers the initial 21 case simulations requested by Nostrum while the second stage focuses on the parametric studies upon the agreement between Nostrum and Michigan Tech. Moreover, the influencing parameters on the post collision angle will be investigated for different injection pressures, chamber pressures as well as for different number of nozzle hole arrangement. This proposed work will be helpful for designing impinging jet injectors and evaluating injector's spray characteristics for Nostrum's novel impinging injector technology.
 
Statement of Work:
The specific works are as follows:
1. Preparing input files for running simulations
- This includes calculating the coordinates and vectors of each nozzle for different multi-hole impinging jet injectors and setup the input files using Converge Studio.
2. Running simulations as per the simulation test matrix
3. Post-processing the simulation output results
- Post-processing of simulation data will be performed using EnSight which is installed in the PI lab.
- New version of EnSight will be purchased based on the progress and loads of the simulated data.
4. Analyze the results and summary
- The post processing will include measuring post collision angle and analyzing the averaged ratio between the post collision angle and collision angle for different conditions as well as exploring the reason behind it.

Awarded Amount: $38,000

Developing a Talent Pipeline: Inspiring Future Naval Engineers and Scientists using Real-World Project Based Instruction

Investigators
Principal Investigator: Andrew Barnard
Co-PI: Nina Mahmoudian
Co-PI: Guy Meadows
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

High BMEP and High Efficiency Micro-Pilot Ignition Natural Gas Engine

Investigators
Principal Investigator: Jeffrey Naber
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Awarded Amount: $137,905

Demonstration of Densification of Biocoal Prepared from Low Lignin Woods

Investigators
Principal Investigator: Ezra Bar-Ziv
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Toward Undersea Persistence

Investigators
Principal Investigator: Nina Mahmoudian
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Evaporation Sub-Model Development for Volume of Fluid (eVOF) Method Applicable to Spray-Wall Interaction Including Film Characteristics with Validation at High Pressure-Temperature Conditions

Investigators
Principal Investigator: Seong-Young Lee
Co-PI: Jeffrey Naber
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Center for Novel High Voltage Temperature Materials and Structures

Investigators
Principal Investigator: Gregory Odegard
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Technical Survey on High Efficient Intensive Cooling Control Technology

Investigators
Principal Investigator: Chang Choi
Co-PI: Jeffrey Allen
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Project Description

Quenching, rapid cooling, has been used to improve hardness and reduce crystallinity by preventing low temperature processes of phase transformations. So cooling alloys and steels in an extremely rapid manner produces martensitic microstructure in their surfaces. Conventional quenching methods use oil, polymer, air, and water. In this proposal, intensive quenching using high velocity water flows is proposed to improve heat-extraction rate by increasing 3-5 times greater heat fluxes from the heated surface of metals. This method is highly efficient and ecofriendly because it uses water and provides greater heat-extraction rates resulting in greater temperature gradient in the sample. This temperature gradient forms compressive stresses from the surface that mainly eliminates cracking. So the intensive quenching keeps the residual surface stresses compressive, while the conventional quenching normally produces tensile or neutral residual surface stresses. The main goal of this project is to establish fundamental and practical technology on intensive quenching heat treatment.

 Michigan Tech will do survey on intensive heat treatment technologies available and/or practical in the world and also do corresponding analytical studies for Year I. For the second year, Michigan Tech will continue doing market survey and analyzing recent research trends for intensive quenching and traditional heat treatment technologies. For Year III, Michigan Tech will provide future market trends and comprehensive technology analysis on heat treatment.

Awarded Amount: $176,724

Center for Novel High Voltage/Temperature Materials and Structures

Investigators
Principal Investigator: Gregory Odegard
Co-PI: Julia King
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Development of Conformable CNG Tanks for Automotive Development

Investigators
Principal Investigator: Gregory Odegard
Co-PI: Jeremy Worm
Co-PI: Jeffrey Naber
Co-PI: Paul Sanders
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

A group of researchers at Michigan Tech and REL propose to design, develop, integrate, and test a CNG tank that will have a conformable shape for efficient storage in a light-duty pick-up truck. Michigan Tech is well known for its research in automotive technology, design optimization via simulation, and aluminum material research. REL is the leader in design and manufacturing of aluminum conformable pressure tanks.

This research will be conducted in two phases. Phase 1 (1 year) will be a proof-of concept phase in which existing technology will be used to fabricate, integrate, and test a conformable tank for a light-duty truck using existing materials technology and fabrication techniques. In Phase 1, the materials development effort will initiate.

The Phase 1 tank is a simple rectangular box geometry to demonstrate capability of non-cylindrical shapes. The end of Phase 2 (2 years) will result in a conformable tank with an optimized internal structure and improved lightweight material for greater efficiency, capacity and durability. The optimized tank will be integrated and tested on a light-duty pick-up truck that has been converted for CNG.

Awarded Amount: $2,107,965

CAREER: Autonomous Underwater Power Distribution System for Continuous Operation

Investigators
Principal Investigator: Nina Mahmoudian
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

CPS: Breakthrough: Toward Revolutionary Algorithms for Cyber-Physical Systems Architecture Optimization

Investigators
Principal Investigator: Ossama Abdelkhalik
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

MicroCSPs Contribution on the Management of an Electrical Grid Including Renewable Energy Sources

Investigators
Principal Investigator: Paul Sanders
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Collaborate with Mohammadia School of Engineering (EMI), Rabat, Morocco within the framework of the Project: "MicroCSPs Contribution on the Management of an Electrical Grid Including Renewable Energy Sources."

 Statement of Work:

  • Support the Design of an intelligent monitoring system for load balancing of a network based on a CSP with storage and photovoltaic panels.
  • Help and support in the study of the integration of CSP in the Moroccan grid.
  • Support the Economic Survey of the implementation of the CSP in the Moroccan power grid in the short term.
  • Support the calculations of the cost of energy generation by the CSP.
  • Support the calculations of an appropriate cost price PPA (Power Purchase Agreement).
  • Transfer of skills where desired.

 

Awarded Amount: $17,616

GOALI: Collaborative Research: Easily Verifiable Controller Design

Investigators
Principal Investigator: Mahdi Shahbakhti
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

MRI: Acquisition of a High Resolution Transmission Electron Microscope for In Situ Microscopy Research and Education

Investigators
Principal Investigator: Reza Shahbazian Yassar
Co-PI: Stephen Hackney
Co-PI: Claudio Mazzoleni
Co-PI: Tolou Shokuhfar
Co-PI: Yoke Khin Yap
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

NRI: Co-Robots to Engage Next Generation of Students in STEM Learning

Investigators
Principal Investigator: Nina Mahmoudian
Co-PI: Michele Miller
Co-PI: Mohammad Rastgaar Aagaah
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Electrospray from Magneto-electrostatic Instabilities

Investigators
Principal Investigator: Lyon King
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Fundamental Understanding on the Role of Structural Defects on Lithiation of Nanoscale Transition Metal Oxides

Investigators
Principal Investigator: Reza Shahbazian-Yassar
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Overview:

Nano-sized transition metal oxides (TMO) are promising materials for lithium-ion batteries. These materials operate through conversion reactions and are associated with much higher energy densities than intercalation reactions. Extensive research is ongoing on the electrochemical characterization of TMO-based electrodes; however, many fundamental questions remained to be addressed. For instance, TMOs exhibit a mysterious extra capacity beyond their theoretical capacity through mechanisms that are still poorly understood. In addition, nano-sized TMOs are highly vulnerable to structural defects produced during synthesis that can alter lithium ion pathways by perturbing the local electronic and lattice strains. No experimental work has been reported to reveal the underlying mechanisms that can correlate structural defects to the electrochemical lithiation in TMOs owing to the difficulty in characterizing structure at the nanoscale, particularly at buried interfaces. This research aims to fill this gap.

The objective is to understand the underlying atomistic mechanisms by which structural defects such as hetrointerfaces, heteroatoms, dislocations, twining, and grain boundaries affect the lithiation behavior of TMOs. In order to meet this objective, single TMO nanowires (NWs) will be subjected to in situ electrochemical lithiation inside high-resolution transmission electron microscope (HRTEM) and aberration-corrected scanning transmission electron microscope (CsSTEM). The in situ electrochemical lithiation will be conducted using state-of-the art scanning tunneling microscope (STM-TEM) and conductive atomic force microscope (cAFM-TEM) holders. This unique combination enables the study of evolution of local lattice strains and electronic perturbations at the vicinity of defects with unprecedented spatial resolutions better than 0.7 A and chemical sensitivity down to 0.35eV.

Intellectual Merit: 

The in situ studies will enable research in three poorly understood fields: (I) the effect of structural defects (twins, dislocations, grain boundaries, and hetrointerfaces) on the nucleation of Li20 and TM particles due to conversion reactions in TMOs; (II) the pinning/unpinning effect of impurities or dopants during grain boundary movement associated with the nucleation of Li20 and TM phases; and (III) the evolution of localized strain and electronic structure at the vicinity of structural defects and their effect on Li-ion pathways The new understanding can facilitate the design of structurally-tailored TMOs for Li-ion battery applications. Furthermore, the experimental methodology and protocols to analyze the in situ data can be extended to other nanomaterials to enable high performance batteries.

Awarded Amount: $445,658

Fundamental Investigations for Very High Heat-Flux Innovative Operations of Milli-Meter Scale Flow Boilers

Investigators
Principal Investigator: Amitabh Narain
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

I/UCRC: Novel High Voltage/Temperature Materials and Structures

Investigators
Principal Investigator: Gregory Odegard
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

CAREER: A New Perspective on Biomineralization in Healthy and Dysfunctional Ferritins

Investigators
Principal Investigator: Tolou Shokuhfar
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Abstract

Overview: Apoferritin is an organic cage that captures the toxic free ferrous ions and transforms them into a ferrihydrite and iron oxide crystalline nanoparticle through a complex biomineralization process (the resulting structural protein is called ferritin). Any dysfunction of ferritin protein can result in iron toxicity, serious illness, chronic diseases, and especially neurological diseases. Dysfunction in ferritin results in the alterations in the biomineralization of the ferritin cores, and therefore, understanding the process of biomineralization within ferritin, is of great importance in the study of neurodegeneration and other chronic diseases. While these unique proteins have been the subject of intense research in biology and chemistry fields due to their importance in many chronic diseases, little effort has been made to unveil the dynamics of such biomineralization processes in liquid conditions. To the Pl's knowledge, there has been no direct evidence at atomic level on how the biomineralization or demineralization inside a ferritin protein progresses over time. This research aims to fill this gap.

The objective of this project is to investigate the in situ crystallization of ferrous ions into crystalline ferrihydrite and iron oxide nanoparticles as well as the demineralization of crystalline core in healthy and dysfunction ferritins in unprecedented resolutions within liquids. In-situ studies conducted inside an atomic resolution aberration-corrected scanning transmission electron microscope (STEM) enabling imaging at resolutions better than 1A. A miniaturized graphene-based electron transparent bio/nano reactor compatible with the microscope chamber is utilized to preserve the liquid environment inside the electron microscope. In this graphene bio/nano reactor, ferrous ions delivered to apoferritins through break down of liposomes acting as reservoirs of irons to trigger the biomineralization within apoferritins cores.

The research is the first atomic resolution study of proteinmediated biomineralization and demineralization within a liquid media and inside a transmission electron microscope. This CAREER research unfolds: (I) The nucleation and growth mechanisms of mineral core (ferrihydrite and iron oxide crystals), (II) the existence and evolution of atomic defects (vacancy, twinning, misorientation boundaries, amorphous regions, etc) during the crystallization, (Ill) the evolution of chemical gradient from surface to core of crystals during the biomineralization, (IV) the mechanisms of demineralization due to iron release, and (V) The atomic-scale morphological and structural differences between a healthy and dysfunctional ferritins.

This research probes the ground rules for ferritin biomineralization with the goal to unveil the fundamental differences with dysfunctional ferritins responsible for neurological diseases. In addition, a new research field for the utilization of bio/nano reactors to image complex biochemical reactions at atomic resolutions will be developed. The CAREER plan will impact the society by integrating multi-disciplinary research with education at all levels while promoting diversity. Graduate and undergraduate students involved with the project will be trained in cross-cutting areas.

Awarded Amount: $554,593

CAREER: Steerable Powered Ankle-foot Prostheses for Increased Mobility in Amputees

Investigators
Principal Investigator: Mohammad Rastgaar Aagaah
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Mass Measurements of an Electrospray Beam from a Single Emitter Ionic Liquid

Investigators
Principal Investigator: Lyon King
Co-PI: Kurt Terhune
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

Distributed Agent-based Management of Agile Microgrids

Investigators
Principal Investigator: Gordon Parker
Co-PI: Laura Brown
Co-PI: Steven Goldsmith
Co-PI: Wayne Weaver
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics

NSF/DOE Advanced Combustion Engines: Ignition and Combustion Characteristics of Transportation Fuels under Lean-Burn Conditions for Advanced Engine Concepts

Investigators
Principal Investigator: Seong-Young Lee
Co-PI: Jaclyn Johnson
College/School: College of Engineering
Department(s): Mechanical Engineering-Engineering Mechanics