1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
|
### **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
|
