DINAST Learning

Engineering Course Tutorials

Tap a course to explore topics, key concepts, and formulas.

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Dynamics
5 topics
AI Tutor ↗ ›
Kinematics of Particles +
  • Position, velocity, acceleration (rectilinear & curvilinear)
  • Projectile motion
  • Normal-tangential (n-t) and polar (r-θ) coordinates
  • Relative motion between particles
Kinetics of Particles +
  • Newton's Second Law: ΣF = ma
  • Work-Energy Theorem: T₁ + ΣU = T₂
  • Impulse-Momentum: mv₁ + ΣImp = mv₂
  • Impact (central & oblique): coefficient of restitution e = (v₂′ − v₁′)/(v₁ − v₂)
Kinematics of Rigid Bodies +
  • Translation, rotation, general plane motion
  • Instantaneous center of zero velocity (IC)
  • Relative velocity: vB = vA + ω × rB/A
  • Relative acceleration: aB = aA + α × rB/A − ω²rB/A
Kinetics of Rigid Bodies +
  • ΣF = mā (force equations)
  • ΣMG = Iα (moment about center of mass)
  • Work-Energy for rigid bodies: T = ½mv² + ½Iω²
  • Angular Impulse-Momentum: HG1 + ΣMG·Δt = HG2
Vibrations +
  • Free undamped: ωn = √(k/m), period T = 2π/ωn
  • Damped free vibration: underdamped, critically damped, overdamped
  • Forced vibration & resonance
  • Logarithmic decrement
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Strength of Materials
7 topics
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Stress & Strain +
  • Normal stress σ = P/A; Shear stress τ = V/A
  • Hooke's Law: σ = Eε; τ = Gγ
  • Poisson's ratio: ν = −εlateral/εaxial
  • Factor of safety: FS = σult / σallow
Axial Load +
  • Deformation: δ = PL/(AE)
  • Statically indeterminate axial members
  • Thermal deformation: δT = αΔTL
  • Stress concentrations: σmax = K·σavg
Torsion +
  • Shear stress: τ = Tc/J
  • Angle of twist: φ = TL/(GJ)
  • Polar moment: J = πd⁴/32 (solid), π(do⁴ − di⁴)/32 (hollow)
  • Power transmission: T = P/ω
Bending +
  • Flexure formula: σ = −My/I
  • Shear formula: τ = VQ/(Ib)
  • Shear & moment diagrams (draw methodically)
  • Moment of inertia: I = bh³/12 (rectangle)
Beam Deflection +
  • Double integration: EI·d²y/dx² = M(x)
  • Superposition — use standard beam tables
  • Boundary conditions: pin (y=0), fixed (y=0, dy/dx=0)
  • Castigliano's theorem: δ = ∂U/∂P
Columns & Buckling +
  • Euler's critical load: Pcr = π²EI/(Le)²
  • Effective length: fixed-free=2L, pin-pin=L, fixed-pin=0.7L, fixed-fixed=0.5L
  • Slenderness ratio: Le/r
  • Short vs. long column cutoff
Combined Loading & Failure Theories +
  • Principal stresses: σ1,2 = (σx+σy)/2 ± √[(σx−σy)²/4 + τ²]
  • Max shear stress: τmax = √[(σx−σy)²/4 + τ²]
  • Von Mises (ductile): σ' = √(σ₁² − σ₁σ₂ + σ₂²) ≤ Sy
  • Max Normal Stress (brittle): σmax ≤ Sut
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Statics
6 topics
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Force Systems +
  • Resultant of concurrent forces: R = ΣF (vector sum)
  • Moment of a force: M = r × F
  • Couple: M = Fd (equal, opposite, non-collinear forces)
  • Equivalent force-couple system
Equilibrium +
  • 2D: ΣFx = 0, ΣFy = 0, ΣM = 0
  • 3D: ΣFx=ΣFy=ΣFz=0, ΣMx=ΣMy=ΣMz=0
  • Free-body diagram (FBD) — draw it every time
  • Support reactions: pin, roller, fixed, link
Trusses +
  • Method of Joints: ΣFx=0, ΣFy=0 at each joint
  • Method of Sections: cut through ≤3 unknown members
  • Zero-force members (visual inspection rules)
  • Simple truss: m = 2j − 3
Frames & Machines +
  • Frames: rigid structure, multi-force members
  • Machines: movable parts, designed to transmit forces
  • Analyze each member as a separate FBD
  • Internal pin forces are equal & opposite on adjacent members
Distributed Loads & Centroids +
  • Resultant of distributed load = area under load curve
  • Acts at centroid of the load diagram
  • Centroid: x̄ = ΣxA/ΣA, ȳ = ΣyA/ΣA
  • Composite shapes — subtract holes
Friction +
  • Static friction: F ≤ μsN
  • Kinetic friction: F = μkN
  • Impending motion: F = μsN (equality holds)
  • Wedge & screw problems: draw FBD for each surface
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Engineering Economy
7 topics
AI Tutor ↗ ›
Time Value of Money +
  • Future value: F = P(1+i)ⁿ → (F/P, i%, n)
  • Present value: P = F(1+i)⁻ⁿ → (P/F, i%, n)
  • Uniform series (annuity): F = A[(1+i)ⁿ−1]/i
  • Gradient series: uniform (G) and geometric (g)
Interest & MARR +
  • Simple interest: I = Pni
  • Compound interest (standard for all problems)
  • Nominal vs. effective rate: ieff = (1 + r/m)m − 1
  • MARR = Minimum Attractive Rate of Return (hurdle rate)
Present Worth (PW) Analysis +
  • PW > 0 → accept; PW < 0 → reject
  • Compare alternatives over same study period
  • LCM of lives for repeated projects
  • Capitalized Cost (CC) = A/i (infinite horizon)
Annual Worth (AW) Analysis +
  • AW = PW × (A/P, i%, n)
  • AW > 0 → accept
  • Capital Recovery: CR = P(A/P,i,n) − SV(A/F,i,n)
  • No LCM needed — AW repeats automatically each cycle
Rate of Return (ROR) +
  • Find i* where PW = 0 (trial & interpolation)
  • IRR > MARR → accept
  • Incremental ROR: Δi* > MARR → choose higher-cost option
  • Multiple IRRs possible — check sign changes in cash flow
Benefit-Cost Analysis +
  • B/C ratio = PW(Benefits) / PW(Costs)
  • B/C ≥ 1.0 → justified
  • Incremental B/C for comparing alternatives
  • Used primarily for public-sector projects
Depreciation & Taxes +
  • Straight-Line (SL): d = (P − SV)/n
  • MACRS — used for US tax purposes (given tables on exam)
  • Book value: BV = P − accumulated depreciation
  • After-tax AW includes tax savings from depreciation