Fundamentals of Transportation/Analogs
Transportation networks have analogs with network processes in other systems, such as water networks, structures, and electrical networks. Some of the relationships are outlined below.
| Concept | Transportation (traffic) | Water (pipe flow / hydraulics) | Structures (linear elastic / statics) | Electrical (DC circuits) |
|---|---|---|---|---|
| Node law (conservation / equilibrium) |
Σqin − Σqout = s (node source/sink s) |
ΣQin − ΣQout = S (node withdrawal/injection S) |
ΣFx = 0, ΣFy = 0, ΣM = 0 |
ΣI = 0 (Kirchhoff’s Current Law, KCL) |
| Element / link law (constitutive) |
Kinematics: q = k·v Link cost (often nonlinear): t = t(q) |
Head/pressure loss: Δp = Rh·Qn (linear case: Q = Ch·Δp) |
Hooke (axial): F = k·δ | Ohm: V = R·I Also: I = G·V, G = 1/R |
| Loop / compatibility law | Wardrop user equilibrium (OD pair): fr > 0 ⇒ cr = c* fr = 0 ⇒ cr ≥ c* |
Loop head balance: ΣΔHℓ = 0 |
Compatibility (parallel members): δ1 = δ2 = … |
ΣV = 0 around any closed loop (Kirchhoff’s Voltage Law, KVL) |
| “Through” variable (adds at node) |
Flow q | Discharge Q | Member force F | Current I |
| “Across” variable (equal in parallel) |
Travel time / cost t or c | Pressure p or head H | Displacement δ | Voltage V |
| Resistance / conductance analog | Increasing cost t(q) (generally nonlinear) Marginal: dt/dq |
Hydraulic resistance Rh Conductance Ch = 1/Rh |
Stiffness k Compliance 1/k |
Resistance R Conductance G = 1/R |
Notes
[edit | edit source]- Hydrostatics (no-flow special case): p = p0 + ρgh (gauge form: p = ρgh).
Symbols
[edit | edit source]Transportation
[edit | edit source]- q = flow (vehicles/time)
- k = density (vehicles/length)
- v = speed (length/time)
- t, c = travel time / generalized cost
- fr = route flow on route r
- cr = cost on route r
- c* = minimum cost for the OD pair (under user equilibrium)
Water (hydraulics)
[edit | edit source]- Q = discharge (volume/time)
- p = pressure
- H = hydraulic head
- ρ = fluid density = m/V
- g = gravitational acceleration
- h = elevation
- Rh = hydraulic resistance
- Ch = hydraulic conductance = 1/Rh
- n = exponent in loss law (model-dependent)
Structures
[edit | edit source]- F = force
- M = moment
- δ = displacement/deflection
- k = stiffness
Electrical
[edit | edit source]- V = voltage
- I = current
- R = resistance
- G = conductance = 1/R