Fundamental Engineering Mechanics and Thermodynamics Formulas
Classified in Physics
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Vector Mechanics and Statics
Scalar product: a · b = |A| |B| · cos(θ)
Projection of vector b onto a: Pa b = (a · b) / |A| · a / |A|
Moment of force about a point: M0 = OA × F
Reduction of Force Systems
Reduction to a new center: Mp = M0 + PO × R (where R is the backbone/resultant force and PO is the vector from the new point to the original center).
Force-couple system: Ftot = Rsystem → M0 = Mpair.
Newton's Laws of Motion
- 1st Law: A particle on which no forces act (or R = 0) will maintain a constant velocity (v = constant).
- 2nd Law: A particle subjected to an experimental force undergoes acceleration.
- 3rd Law: If body A exerts a force on body B, body B returns a force of the same magnitude and opposite direction.
Classification of Unranked Force Systems
| Condition: M · R ≠ 0 | Condition: M · R = 0 |
|---|---|
| R ≠ 0, M ≠ 0: System reduces to a torque/force (wrench) on the central axis. | R = 0, M ≠ 0: Pure torque/couple at any point in space. |
| R ≠ 0, M = 0: System reduces to a single force (sliding vector) on the central axis. | R = 0, M = 0: Null system. |
Constraints and Degrees of Freedom
Types of Constraints
- Simple (1 degree): Simple support, rod, or wire.
- Double (2 degrees): Joint or rigid guideway.
- Triple (3 degrees): Fixed support (underrun) or welding.
Stability and Determinacy
- External stability (Sustentation): Ge = 3 - C
- Internal stability (Constitutional): Gi = 3N - 3 - C
- Global stability: G = Ge - Gi
- Classification: Isostatic (G = 0), Mechanism (G > 0), or Hyperstatic (G < 0).
Friction and Fluid Mechanics
Sliding friction: Fr < Fr,max = μ × N. For a shift, x = 0 (related to the normal force position).
Hydrostatics
Pressure at a point at rest: P = PATM + ρgh
Forces on Walls
- Horizontal wall: F = (PATM + ρgh) · S
- Slanting wall: F = (ρ · g · lx · h2) / (2 · sin θ)
Fluid Dynamics
Flow rate: G = V · S
Reynolds Number: Re = ρvD / μ
Energy Losses
- Localized losses: hl = K · (v2 / 2g)
- Viscosity losses: h = f · (L / D) · (v2 / 2g)
Bernoulli Equation: P1 / ρg + z1 + v12 / 2g + hPump = P2 / ρg + z2 + v22 / 2g + Losses + hTurbine
Note: Multiply the kinetic energy term by 2 for laminar flow and by 1 for turbulent flow.
Thermodynamics and Heat Transfer
Thermal Expansion and Stress
- Linear expansion: ΔL = α L0 Δt
- Surface expansion: ΔA = 2α A0 Δt
- Cubic expansion: ΔV = 3α V0 Δt
- Linear thermal stress: F / S = α E Δt
- Volumetric thermal stress: Δp = 3α B Δt
Heat Transfer Mechanisms
- Sensible heat: Q = m · cp (t2 - t1)
- Latent heat: Q = m · L
- Conduction heat transfer: Q = (t1 - t2) / Req
Thermal Resistance (Reb)
- Flat wall: Reb = D / (K · S)
- Cylindrical pipe: Reb = [1 / (2πLK)] · ln(r2 / r1)
Association of Conductors
- Series (Composite walls): Req = Σ Ri
- Parallel: Req = (Σ Ri-1)-1