### **Module 2: Algorithm Developments for Mechanical Engineering**

* **Numerical Methods**
  1. Beam deflection solved with `numpy.linalg.solve` (Statics/Structures)
  2. Cooling of a hot plate with `scipy.integrate.odeint` (Thermal sciences)

* **Version Control**
  1. Git workflow on a bike frame design project (Solid mechanics / FEA data)
  2. Collaborative control system simulation (PID tuning in Python)

* **Problem Solving Strategies**
  1. Break down multi-step dynamics problem: projectile with drag (Dynamics/Fluids)
  2. Decompose Rankine cycle analysis into modular Python functions (Thermo/Power cycles)

* **Code Documentation**
  1. Document a function that computes stress concentration factor (Solid mechanics)
  2. Document a PID control simulation for DC motor speed (Mechatronics/Controls)

* **Code Libraries & Resources**
  1. Use `matplotlib` + `numpy` for plotting vibration response of a 2-DOF spring-mass system (Dynamics)
  2. Use `CoolProp` for real gas property lookup (Thermo/Fluids)

* **AI-Assisted Programming**
  1. Have AI draft code for a 1D heat conduction simulation, then debug/validate (Heat Transfer)
  2. Generate starter code for stress-strain curve fitting, then refine it (Solid Mechanics)

* **Verification & Validation**
  1. Compare hand solution vs. Python solution of a static truss (Statics/Structures)
  2. Validate simulated temperature profile in a fin against analytical solution (Heat Transfer)

* **Error**
  1. Show truncation error in numerical derivative of displacement data (Dynamics experiment)
  2. Compare Simpson’s rule vs trapezoidal integration for work done in P–V diagram (Thermo)

### **Module 3: Applications of Computational Mathematics in Mechanical Engineering**

#### **Linear Equations**

* **Lecture: Linear Equations**
  1. Solve a 3-bar truss reaction forces system (Statics/Structures)
  2. Solve nodal voltages in a resistive electrical network (Mechatronics)

* **Lecture: Linear Algebra**
  1. Stress transformation with rotation matrices (Solid Mechanics)
  2. Velocity transformation in 2D robotic arm kinematics (Mechatronics/Robotics)

* **Lecture: LU Decomposition**
  1. Discretized fin heat conduction (Heat Transfer)
  2. 1D beam bending using finite difference discretization (Simple FEA)

#### **Non-Linear Equations**

* **Lecture: Bracketing Methods**
  1. Equilibrium temperature with radiation + convection balance (Heat Transfer)
  2. Solve spring-mass system with nonlinear stiffness (Solid Mechanics)

* **Lecture: Open Methods**
  1. Mach number from nozzle area ratio (Aerospace/Compressible Flow)
  2. Natural frequency from transcendental vibration equation (Dynamics)

* **Lecture: Systems of Nonlinear Equations I**
  1. Chemical equilibrium composition (Thermo)
  2. Solve coupled force/displacement in nonlinear truss (Structures)

* **Lecture: Systems of Nonlinear Equations II**
  1. Lift & drag coefficients from nonlinear aerodynamic model (Aerospace)
  2. Nonlinear motor torque & current equations (Mechatronics/Controls)

#### **Numerical Differentiation & Integration**
* **Lecture: Differentiation**
  1. Estimate velocity/acceleration from piston displacement data (Dynamics)
  2. Approximate strain rate from stress–strain curve (Solid Mechanics)

* **Lecture: Integration**
  1. Work from P–V data (Thermo)
  2. Area under vibration response curve for damping energy loss (Dynamics/Controls)

#### **ODEs**

* **Lecture: Explicit Methods**
  1. Cooling of hot sphere (Newton’s law of cooling) (Heat Transfer)
  2. Pendulum motion (small vs. large angle) (Dynamics)

* **Lecture: Implicit Methods**
  1. Transient 1D heat conduction (Heat Transfer)
  2. Mass-spring-damper under step input (Dynamics/Controls)

* **Lecture: Systems of ODEs**
  1. Coupled tanks draining problem (Fluids)
  2. 2-DOF vibration system (Solid Mechanics/Dynamics)

#### **PDEs**
* **Lecture: Elliptic PDEs (Finite Difference)**
  1. 2D steady-state conduction in a plate (Heat Transfer)
  2. Deflection of a membrane under load (Structures/FEA)

* **Lecture: Parabolic/Hyperbolic PDEs (Finite Difference)**
  1. 1D transient conduction (Heat Transfer)
  2. Vibrating string or beam (Dynamics/Aerospace structures)


* Statics/Structures → Truss, beams, stresses
* Dynamics → Vibrations, pendulum, projectile, damping
* Solid Mechanics → Stress/strain, nonlinear springs
* Thermo/Fluids/Heat → Cooling, P–V, conduction, radiation
* FEA → simple discretized beams/fins/plates
* Mechatronics/Controls → motor dynamics, robotic arms, PID