--- title: "Static vs dynamic mathematical model" subject: "Simulation and Modeling" chapter: 2 key: "semester_8/simulation_and_modeling:static-vs-dynamic-math-model" source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/static-vs-dynamic-math-model site: BCT Engineering --- # Static vs dynamic mathematical model Simulation and Modeling · Chapter 2: Physical and Mathematical models - Asked in: 2074 Bhadra, 6 marks; 2073 Bhadra, 6 marks; 2069 Bhadra (old course), 10 marks (asked 3 times) - Video: [System Dynamics and Control: Module 2c - Static vs. Dynamic Models](https://www.youtube.com/watch?v=orKUimzyxYw) — Rick Hill - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/static-vs-dynamic-math-model - Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-2 ## Question Differentiate between static mathematical model and dynamic mathematical model with suitable examples. ## Short answer Static models describe a system at a fixed instant or equilibrium without time, e.g., Ohm's law V=IR. Dynamic models incorporate time, using differential/difference equations to describe state evolution, e.g., population growth dP/dt=kP. Static models give instantaneous relationships; dynamic models capture transient behaviour and depend on history. ## Answer A **static mathematical model** describes the relationship between variables at a fixed moment or in equilibrium; time does not appear in it. A **dynamic mathematical model** describes how the variables change with time, normally using differential or difference equations. | Point | Static model | Dynamic model | |---|---|---| | Time | Not a variable | Time is an independent variable | | Form | Algebraic equations | Differential or difference equations | | Describes | System at one instant or steady state | System behaviour over time, transient and steady | | Output | Depends only on the present input | Depends on present input and past history/state | | Memory | None | Has memory (state variables) | | Solution | Direct substitution or algebra | Analytical integration or numerical methods, simulation | | Computation | Simple | More complex | | Use | Design, equilibrium and sizing studies | Prediction, stability and response studies | #### Examples of static models - Ohm's law, `V = I R`: the voltage depends on the present current only. - Straight-line cost model: `C = F + v x`, where F is the fixed cost, v the unit cost and x the quantity. - Economic order quantity: `Q = sqrt(2 D S / H)`. - Population of a town estimated by `P = density x area`. #### Examples of dynamic models - Population growth: `dP/dt = k P`, solution `P = P0 e^(kt)`. - Spring-mass-damper (car suspension): `M x'' + D x' + K x = F(t)`. - RC circuit: `R C dv/dt + v = V(t)`, whose capacitor voltage changes with time. - Inventory over days: `I(n+1) = I(n) + production(n) - demand(n)` (difference equation, a discrete dynamic model). A dynamic model reduces to a static one when all derivatives are set to zero (steady state). For example, setting `dv/dt = 0` in the RC circuit gives `v = V`, a static relationship. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the main difference between static and dynamic mathematical models? A. Static models use differential equations B. Dynamic models do not need initial conditions C. Both have time as an independent variable D. Time is not a variable in a static model but is in a dynamic model Answer: D. A static model describes a fixed moment or equilibrium with no time; a dynamic model describes change with time. (Key point: Role of time) 2. Which statement about output is correct? A. A static model's output depends only on the present input; a dynamic model's also depends on past history/state B. A static model's output depends on past history C. Dynamic model output depends only on present input D. Neither model depends on input Answer: A. Static models have no memory; dynamic models have state variables that carry history. (Key point: Memory) 3. Which is a dynamic model? A. V = I R B. Q = sqrt(2 D S / H) C. M x'' + D x' + K x = F(t) D. C = F + v x Answer: C. The spring-mass-damper equation contains derivatives with respect to time; the others are algebraic. (Key point: Classifying examples) 4. In the RC circuit R C dv/dt + v = V(t), what do you get by setting dv/dt = 0? A. v = 0 B. v = V, a static relationship C. v = V/RC D. The model becomes unsolvable Answer: B. At steady state all derivatives are zero, so the dynamic model reduces to the static relationship v = V. 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