Mechanical Engineering Fundamentals and Applied Mathematics
The fundamentals block of the Iran Nezam mechanical design exam tests the arithmetic layer that every later domain silently assumes: unit discipline, mathematical modelling, and clean numerical work. Questions here are short but unforgiving - one unit slip converts a correct method into a wrong answer, and there is rarely part credit.
What is tested
- Unit conversions: 1 bar = 100 kPa = 14.504 psi; 1 kgf = 9.80665 N; 1 kcal = 4.1868 kJ; 1 hp = 0.7457 kW; T(K) = t(°C) + 273.15.
- Differential equations: separable and linear first-order ODEs, Laplace transforms for step and ramp inputs.
- Linear algebra: 2×2 and 3×3 systems, determinants, eigenvalues used in vibration and buckling.
- Numerical methods: Newton–Raphson iteration, trapezoidal and Simpson integration, convergence criteria.
- Statistics: mean and standard deviation, normal distribution, tolerance stack-up (worst case versus root-sum-square).
Worked example
Solve y′ + 2y = 4 with y(0) = 0. Homogeneous part: y_h = Ce^(−2t). A constant particular solution y_p = 2 satisfies 0 + 2(2) = 4, so y = Ce^(−2t) + 2. Applying y(0) = 0 gives C = −2, hence y = 2(1 − e^(−2t)). The time constant is 1/2 s, and the response reaches 63.2% of its final value (1.264) at t = 0.5 s.
Traps
- Mixing gauge and absolute pressure inside gas-law and compressor calculations.
- Entering degrees instead of radians in series expansions and derivatives of trigonometric terms.
- Quoting more significant figures than the input data supports.
- Assuming RSS tolerance stack-up equals the worst-case sum; it is always smaller.