**Applied Mechanics **

The imperative of energy efficiency continues to be the driving force behind numerous developments in mechanical engineering, from new materials to airplane rotor blade design. The Applied Mechanics option in Mathematics and Engineering blends elements of traditional Mechanical Engineering with the strong mathematics foundation that all students in the program receive.

Students take engineering courses in subjects such as classical mechanics, fluid mechanics, and thermodynamics, and specialized courses in engineering systems, control theory, and analytical mechanics. Graduates of this option are exceptionally well-prepared for careers in the traditional Mechanical Engineering fields as well as for careers in any field requiring strong analytical skills.

**Second Year Courses**

MECH 221: Solid Mechanics IReview of statics, forces and equilibrium, internal forces in simple structures and other material from first year. Further development of axial, torsion, shear and bending moment diagrams, and concepts of stress and strain. Introduction to mechanical properties of materials, centroids and moments of areas, axial stress, flexural stress, transverse shear stress, calculation of displacement by integration, combined loading, and stress transformation. This course is designed primarily for mechanical engineering students.

MECH 210: Electronic Circuits and Motors for MechatronicsThis introductory course for mechanical engineering students begins with a review of the concepts of resistance, capacitance, and inductance. Circuit analysis techniques are then applied to characterize the behaviour of commonly used mechatronic circuits including devices such as transformers, diodes, solenoids, DC motors and actuators. Transistors are introduced in switching applications. Selection and testing of electric motors and drivers/controllers for stationary and mobile mechanical applications. Lab activities will focus on design, construction, and testing of microcontroller based mechatronic systems for practical applications. Students will solve mechatronics problems analytically and computationally in an active learning, tutorial environment.

MREN 230: Thermodynamics and Heat TransferThis course introduces fundamental thermodynamics and heat transfer concepts needed to analyze thermal systems including: ideal gas laws; work and heat; conservation of energy; thermodynamic properties of pure substances; equations of state; applications to open and closed systems; heat transfer by conduction, convection and radiation. Theory will be complemented with a series of labs that introduce temperature measurement devices and thermal circuit analysis.

MREN 241: Fluid Mechanics and Fluid PowerAn introductory course in fluid mechanics with application to fluid power systems. Topics include properties of fluids, fluids at rest, dimensional analysis, the laws of conservation of mass and momentum, Bernoulli’s equation for incompressible flow and the energy equation, flow measurements, elementary pipe flow problems including losses due to pumps, valves etc. Laboratories will introduce students to pressure and flow measuring devices, pneumatic and hydraulic components and actuators, and circuit analysis of fluid power systems

ENPH 225: MechanicsExtension of classical mechanics and engineering applications. Plane dynamics, relative motion and forces in moving and accelerated reference frames. Introduction to general three-dimensional motion of a rigid body, inertia tensor and steady-state precession. The laws of conservation of mass, momentum and energy.

**Third Year Courses**

MECH 321: Solid Mechanics IIThis course continues the study of solid mechanics. On completion of the course students will be able to: Calculate the total normal and shear stress at a point and sketch the stress distributions on a cross-section of a structural component (such as a crank) experiencing 3D combined (axial, transverse and/or moment causing) loads and non-symmetric loads; Calculate the residual normal or shear stress at a point and sketch the stress distribution on a cross-section of a structural component that is experiencing axial, torsional and/or bending loads followed by unloading; Calculate the normal or shear stress at a point on a cross-section of a structural component that is under load (axial, torsional and/or bending) and is supported in a statically indeterminate configuration (using force balance equations together with compatibility equations derived from known boundary conditions); Calculate the normal or shear stress at a point on a cross-section of a structural component that is under load (axial, torsional and/or bending) and contains one or more locations of stress concentration; Calculate, using general equations and/or graphically using a Mohr’s circle, the normal and shear stress and/or strain transformations at a point within a structural component under load as a function of the orientation relative to a fixed coordinate system and find the maximum in-plane normal and shear stress and/or strain; Calculate the deflections and angles of deflection at any point on a transversely loaded beam of uniform cross-section using the principle of superposition and the standard equations for single loads acting on simply supported beams; Solve for critical loads in terms of buckling for concentrically and eccentrically loaded columns; Calculate the optimum dimensions (design) for shafts and beams under combined 3D loading based on specified material failure criteria; Design mechanism or structural components to withstand all forces for given loads, maximum deflection tolerances, factor of safety and material properties.

MECH 328: Dynamics and VibrationThis course covers the kinematics and dynamics of rigid bodies in two and three dimensions, as well as an introduction to vibrations. Topics in dynamics include: mathematically rigorous kinematic analysis, Newton’s laws, energy methods, impulse and momentum methods, mass moments of inertia, and gyroscopic motion. Topics in vibrations include: free and forced vibration of single-degree-of-freedom systems, undamped and damped systems, equivalent single degree of freedom system of continuous elements/systems using energy equivalence and equation of motion.

MECH 330: Applied Thermodynamics IIA continuation of MECH 230 with selected topics such as gas and vapour power cycles, refrigeration, mixtures of gases and vapours, combustion and available energy.

MECH 398: Mechanical Engineering Laboratory IThis is the first of two laboratory courses in the third year of the General Option of the Mechanical Engineering program. Lecture topics and course assignments are selected to provide the background required to undertake the laboratory work.

MECH 323: Machine Design IThis course emphasizes the application of theoretical and engineering background taught in other courses, but also relies heavily on empirical approaches and simplifications of theory. Core material includes static and fatigue failure theories and the design/specification of selected machine elements. The course is centered around a major design project which is undertaken in groups.

MECH 341: Fluid Mechanics IIA second course in fluid mechanics covering the differential form of conservation laws, boundary layer and external flows, compressible flows and the operation of rotational fluid machinery. On completion of the course students will be able to: Apply control volume analysis to mass, momentum and energy conservation; Apply differential form of mass and momentum conservation to the concept of flow field and its properties, including Navier Stokes equations; Apply stream function and velocity potential to the analysis of two-dimensional inviscid flows, and use the superposition principle to build complex flow fields from building block ingredients; Calculate drag and lift on solid bodies such as airfoils; Explain boundary layer flows, including the concept of various boundary layer thicknesses, shape factor, flow separation and the difference between laminar and turbulent boundary layers; Explain compressible flow features based on one-dimensional compressible subsonic and supersonic flows, with and without normal shock waves; Calculate design parameters of rotational fluid machinery, including centrifugal pumps and wind turbines.

MECH 399: Mechanical Engineering Laboratory IIThis is the second of two laboratory courses in the third year of the General Option of the Mechanical Engineering program. Lecture topics and course assignments are selected to provide the background required to undertake the laboratory work.

**Fourth Year Courses**

MTHE 430: Modern Control TheoryThis course covers core topics in modern control theory: Linearization, existence and uniqueness of trajectories for nonlinear and linear systems, the transition matrix, controllability, observabiity, minimal realizations, feedback stabilization, linear state observers, optimal control theory, the linear quadratic regulator, dynamic programming.

MTHE 433: Continuum Mechanics with ApplicationsContinuum mechanics lays the foundations for the study of the mechanical behavior of solids and fluids. After a review of vector and tensor analysis, the kinematics of continua are introduced. Emphasis is given to the concepts of stress, strain and deformation. The fundamental laws of conservation of mass, balances of (linear and angular) momentum and energy are presented together with the constitutive models. Applications of these models are given in the theory of linearized elasticity and fluid dynamics.

**Technical Courses**

In the fourth year of the program, students choose four technical electives to compliment the core courses and explore their individual interests. At least one must be chosen from List I and the remaining from List I or List II so that they satisfy certain engineering design and engineering science criteria. Course offerings change year to year.

See the current Academic Calendar for a complete list of offered technical electives.

### List I

MTHE 353: Probability IIIntermediate probability theory as a basis for further study in mathematical statistics and stochastic processes; probability measures, expectations; modes of convergence of sequences of random variables; conditional expectations; independent systems of random variables; Gaussian systems; characteristic functions; Law of large numbers, Central limit theory; some notions of dependence.

MTHE 406: Introduction to Coding TheoryConstruction and properties of finite fields. Polynomials, vector spaces, block codes over finite fields. Hamming distance and other code parameters. Bounds relating code parameters. Cyclic codes and their structure as ideals. Weight distribution. Special codes and their relation to designs and projective planes. Decoding algorithms.

MTHE 418: Number Theory & Cryptography

Time estimates for arithmetic and elementary number theory algorithms (division algorithm, Euclidean algorithm, congruences), modular arithmetic, finite fields, quadratic residues. Simple cryptographic systems; public key, RSA. Primality and factoring: pseudoprimes, Pollard’s rho-method, index calculus. Elliptic curve cryptography.

MTHE 434: Number Theory & Cryptography

Theory of convex sets and functions; separation theorems; primal-dual properties; geometric treatment of optimization problems; algorithmic procedures for solving constrained optimization programs; applications of optimization theory to machine learning.

MTHE 472: Control of Stochastic SystemsThis course concerns the optimization, control, and stabilization of dynamical systems under probabilistic uncertainty with applications in engineering systems and applied mathematics. Topics include: controlled and control-free Markov chains and stochastic stability; martingale methods for stability and stochastic learning; dynamic programming and optimal control for finite horizons, infinite horizons, and average cost problems; partially observed models, non-linear filtering and Kalman Filtering; linear programming and numerical methods; reinforcement learning and stochastic approximation methods; decentralized stochastic control, and continuous-time stochastic control.

MTHE 437: Topics In Applied Mathematics

Topic: An Introduction to Stochastic Differential Equations (with Applications to Mathematical Finance and Engineering) The aim of this course is to provide a rigorous introduction to the theory of stochastic calculus and stochastic differential equations, and to survey some of its most important applications, especially in Mathematical Finance. The Itô stochastic integral and its associated “Itô Calculus” will be derived in the general framework of continuous semimartingales, leading to a detailed treatment of stochastic differential equations (SDEs) and their properties. The theory thus developed will be applied to selected problems in Mathematical Finance (option pricing and hedging, trading strategies and arbitrage) and Engineering (boundary-value problems, filtering, optimal control). Numerical aspects of SDEs will also be discussed.

MTHE 439: Lagrangian Mechcanics, Dynamics ControlGeometric modelling, including configuration space, tangent bundle, kinetic energy, inertia, and force. Euler-Lagrange equations using affine connections. The last part of the course develops one of the following three applications: mechanical systems with nonholonomic constraints; control theory for mechanical systems; equilibria and stability.

MTHE 457: Statistical Learning

Introduction to the theory and application of statistical algorithms. Topics include classification, smoothing, model selection, optimization, sampling, supervised and unsupervised learning.

### List II

MECH 346: Heat TransferAn introductory course which covers conduction, convection and radiation modes of heat transfer. Both analytical and numerical analysis will be discussed, and concepts will be reinforced through tutorial and laboratory sessions. Latter topics will include combined modes of heat transfer and the design of heat exchangers.

MECH 420: VibrationsConsiders mechanical vibration, the problems it presents and the means of dealing with it. Completes the treatment of systems with two degrees-of-freedom (introduced in MECH 328) and proceeds to systems with higher number of degrees-of-freedom. Co-ordinate systems, types of coupling, matrix formulation, vibration absorbers and dampers, specific and hysteretic damping, Rayleigh’s method, torsional vibration, Holzer method, introduction to the finite element method, beam vibration.

MECH 424: Sustainable Product DesignThis course deals with sustainable product design and manufacture. Topics include: product Life Cycle Analysis issues; Streamlined Life Cycle Analysis and international Life Cycle Analysis standards; Energy, Global Warming Potential, Green House Gas and carbon emission issues (including energy needs in product design and manufacturing); Carbon footprint, basic chemistry of carbon emissions, international standards for carbon emission signatures. Design topics include: product design for manufacture and assembly, design for disassembly and design for environment. Product end-of-life considerations include: recycling, remanufacture and reuse. Students will complete several open ended projects. Guest speakers will be included where possible.

MECH 435: Internal Combustion EnginesThis course covers all aspects of the design and operation of internal combustion engines. Principles of thermodynamics and fluid mechanics are used in the analysis of internal combustion engines. Course content includes discussions on both spark ignition and compression ignition (diesel) engines with special emphasis placed on new engine technologies. Intake, in-cylinder and exhaust flows are considered along with various aspects of combustion phenomenon relevant to engines. This course includes a laboratory involving engine performance measurements made using a dynamometer.

MECH 439: TurbomachineryFluid mechanics and thermodynamics applied to turbomachines; dimensionless performance characteristics; momentum and energy equations; thermodynamics and efficiencies; cascade aerodynamics; compressors and turbines, reaction and stage loading; radial equilibrium; radial flow machines; application of generalized performance to choice of compressors; mechanical details and auxiliary systems.

MECH 441: Fluid Mechanics IIITopics will include: Derivation of equations of motion for incompressible fluids; exact solutions for laminar flows; stability and transition; introduction to turbulence, including turbulent boundary layers, jets, wakes and mixing layers; drag reduction; introduction to the modelling of turbulence.

MECH 444: Computational Fluid DynamicsThis course provides an overview of, and hands-on experience in, the numerical modelling of fluid flows. Finite volume, finite difference and finite elements methods are introduced. Students are expected to gain critical insight into the capabilities and limitations of fluid flow models by numerically simulating various engineering flows and by doing a term project. Topics include: comparison of numerical, experimental and analytical methods in fluid mechanics, numerical grids and their generation, flow equations and their discretization, solution techniques, turbulence modelling and data presentation. Features of commercial codes are critically reviewed.

MECH 448: Compressible Fluid FlowIntroduction and review of work done in earlier courses; basic equations for one-dimensional compressible flow; isentropic one-dimensional flow; steady and unsteady normal shock waves; oblique shock waves; steady and unsteady expansion waves; two-dimensional isentropic flow; nozzle flows; effects of friction and heat transfer; boundary layer flow; design of aircraft engine intake systems; design of supersonic wind-tunnels and shock tubes. Students are expected to have knowledge of fluid mechanics typically acquired in MECH 241/MECH 341. Those who have not taken these or similar courses will need to prepare through self study.

MECH 452: Mechatronics EngineeringThis is a course in mechatronic systems design. Mechatronics Engineering, an integration of computer, electrical and mechanical engineering, is studied in a series of workshops that focus on electronics, microcontrollers, programmable logic controllers and mobile robots. The lectures provide the theoretical background to the workshops, and include discussion of related industrial and commercial applications. The knowledge and experience gained in the lectures and workshops is applied to a team design project. Students will use their knowledge of electric circuits, microcontrollers and control systems typically acquired in MECH 210, MECH 217 and MECH 350.

MECH 455: Computer Integrated ManufacturingThe course will focus on the integration of facilities (machine tools, robotics) and the automation protocols required in the implementation of computer integrated manufacturing. Specific concepts addressed include flexible manufacturing systems (FMS); interfaces between computer aided design and computer aided manufacturing systems; islands of automation.

MECH 456: Introduction To RoboticsThis course will cover the following topics in the field of robotics: historical development; robot components (sensors, actuators, and end effectors, and their selection criteria); basic categories of robots (serial and parallel manipulators, mobile robots); mobility/constraint analysis; workspace analysis; rigid body kinematics (homogeneous transformation, angle and axis of rotation, Euler angles); manipulator kinematics and motion trajectories (displacement and velocity analyses, differential relations, Jacobian matrix); non redundant and redundant sensing/actuation of manipulators; manipulator statics (force and stiffness); singularities; and manipulator dynamics.

ELEC 448: Introduction to Robotics - Mechanics and ControlRobotics is an interdisciplinary subject concerning areas of mechanics, electronics, information theory, control systems and automation. This course provides an introduction to robotics and covers fundamental aspects of modeling and control of robot manipulators. Topics include history and application of robotics in industry, rigid body kinematics, manipulator forward, inverse and differential kinematics, workspace, singularity, redundancy, manipulator dynamics, trajectory generation, actuators, sensors, and manipulator position and contact force control strategies. Applications studied using MATLAB/Simulink software simulation and laboratory experiments.

MECH 465: Computer-Aided DesignConcept of computational design including the choice of the objective function, equality and inequality constraints, and analysis methods; one-dimensional search methods, sensitivity analysis, and the steepest descent method. The principles of the finite element method and its application to stress analysis of mechanical components. The prerequisite may be waived for students with a strong background in solid mechanics from other courses.

MECH 480: Airplane Aerodynamics and PerformanceA technical course on the principles of flight. Techniques for the quantitative prediction of the aerodynamic characteristics of the wing will be described. Extensions to account for real-world effects will be discussed. These results will be used to predict the airplane performance (range, climb rate, maximum speed, etc.). The concept aerodynamic stability will be introduced and discussed. Students are expected to know MATLAB proficiently and have fluids knowledge typically acquired in MECH 241 and MECH 341. Those who have not taken these or similar fluids courses will need to prepare through self study.

MECH 482: Noise ControlAn introduction to the principles of noise control. Topics include: basic properties of sound and noise, the measurement of noise, effects of noise on people, description of sound fields, acoustics of rooms and enclosures, acoustical materials and structures, and noise source identification. A coherent approach to the solution of noise control problems is stressed throughout the course.

MECH 492: Biological Fluid DynamicsAn introductory course on biological flows across a broad range of scales from flagellar motility to the beating heart. Topics range from the dynamics of classical biomedical flows such as the circulatory and respiratory systems. (e.g. wall compliance, pulsatility, and transition to turbulence) through to cellular-level motility and biopropulsion in general over a range of Reynolds numbers. Topics relating to comparative biology (e.g. allometry and evolutionary convergence) and common imaging techniques used for biological flows (e.g. acoustic, nuclear magnetic resonance, optical and x-ray techniques) will be covered as well. (0/0/0/42/0) Students are expected to have sufficient experience with measurements Science and fluid dynamics theory equivalent to MECH 217, MECH 241, and MECH 341, respectively. Those who have not taken such courses will need to prepare through self-study.

MECH 494: Kinematics Of Human MotionIn this course students will explore the application of classical mechanics to the analysis of human motion related to athletics, orthopaedics, and rehabilitation. The course covers the structure of human joints, including experimental and analytical techniques in the study of human joint kinematics; applications to the design of artificial joints and to clinical diagnosis and treatments. Students are introduced to the motion capabilities of the human body and how to develop and study kinematic models of the individual joints of the human body. Experimental methods used to collect kinematic data will be studied through interactive labs. Topics include defining body position and displacement, three dimensional representation of human motion, basic functional anatomy of individual joints, rigid body kinematics (homogeneous transformations, Euler angles, helical axis), intrajoint kinematics, joint modelling, articular surface motion. Three-dimensional kinematics of individual joints is emphasized from the perspective of total joint replacement design.

MECH 495: Ergonomics and DesignThis course provides an overview of ergonomic problems that are addressed in engineering design; including biomechanical, physical and physiological issues. Case studies will range from the design of vehicle cockpits to process control rooms, from industrial manual materials handling tasks to human directed robots, and from domestic tools to biomechanical devices.

MECH 496: Musculoskeletal BiomechanicsDevelops approaches to musculoskeletal biomechanics, including experimental and analytical approaches to movement analysis, experimental instrumentation and devices, and biomechanical devices for musculoskeletal disorder rehabilitations. Analysis of the contribution of external loading, forces generated by muscles and constraints provided by other musculoskeletal structures to predict forces and stresses in musculoskeletal joints and tissues. Numerical and modelling approaches, including inverse dynamics, and optimization, and determination of segmental inertial properties. Biomechanical devices including upper limb and lower limb orthotics and prosthetics. Applications in orthopaedic engineering, movement assessment, ergonomics, joint injury and replacements, and biomechanical system design. Application of machine learning in biomechanics and human movement analysis.

MECH 430: Thermal Systems DesignThis course is concerned with the technical, economic and environmental aspects of conventional and novel methods of energy supply and use. Emphasis will be placed on the analysis and design of thermal systems. Topics include: electric utility demand and supply; the analysis of thermal power generation systems including combined cycle and cogeneration plants; emission control; alternative energy systems. A group project related to the design of a thermal system will form a significant portion of this course. NOTE: Limited enrollment.

MINE 472: Mining Systems, Automation, and RoboticsIn order to address issues related to safety, productivity, and remote operations, the world’s mineral resources industry has been gradually shifting towards the increased use of automated systems and robotically enhanced machines. It is important, therefore, that graduate engineers understand how these new technologies work so as to improve and make best use of them. This online course introduces senior students to the fundamental tools and techniques of automation and robotics as applied to modern mining practice. This course provides an introduction to the basics of systems control, examples of how methods of automatic control can be applied to mining equipment and associated industrial vehicles, as well as to the fundamentals of sensing and navigation as applied to the design of robotic mobile equipment.