---
title: "Dynamic physical model with example"
subject: "Simulation and Modeling"
chapter: 2
key: "semester_8/simulation_and_modeling:dynamic-physical-model"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/dynamic-physical-model
site: BCT Engineering
---

# Dynamic physical model with example

Simulation and Modeling · Chapter 2: Physical and Mathematical models

- Asked in: 2074 Magh, 6 marks; 2073 Magh, 6 marks; 2071 Bhadra, 6 marks (asked 3 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/dynamic-physical-model
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-2

## Question

Explain the dynamic physical model with example (necessary equations and diagrams).

## Short answer

A dynamic physical model is a hardware model built from real parts that imitates time behaviour. Example: automobile suspension modeled by M x'' + D x' + K x = F(t), with mass M, damping D, stiffness K, and force F(t). The response depends on D² versus 4KM.

## Answer

A **dynamic physical model** is a physical (hardware) model whose behaviour changes with time and is used to imitate the time behaviour of a real system. Because it has a physical form, it is built from real parts (mechanical, electrical or fluid) and observed while running. Examples: a wind tunnel with a scale aircraft, a ship model in a water tank, a flight simulator, a mechanical model of a car suspension.

#### Example: automobile suspension (spring-mass-damper)
The car body of mass M rests on a spring (stiffness K) and a shock absorber (damping coefficient D). A force F(t) from the road acts on the body.

```text
      F(t)
        |
     +-----+
     |  M  |   body mass
     +-----+
     |     |
    K (spring)  D (damper)
     |     |
 ====================  road / wheel
```

Newton's second law gives the equation of motion, with x the displacement from equilibrium:

`M x'' + D x' + K x = F(t)`

Here `M x''` is the inertia force, `D x'` the damping force and `K x` the spring force.

- If `D^2 < 4 K M`, the response is oscillatory and slowly dying out (under-damped).
- If `D^2 = 4 K M`, it is critically damped (fastest return without oscillation).
- If `D^2 > 4 K M`, it is over-damped (slow return).

#### Electrical analogue
The same behaviour can be reproduced by a series RLC circuit, `L q'' + R q' + q/C = V(t)`.

| Mechanical | Electrical |
|---|---|
| Mass M | Inductance L |
| Damping D | Resistance R |
| Spring constant K | 1/C (C = capacitance) |
| Displacement x | Charge q |
| Force F | Voltage V |

#### Features
- The model shows real, physical behaviour, so assumptions are fewer.
- It runs in real time (or scaled time) and may be costly to build and modify.
- Parameters (D, K) can be changed to find the best design before manufacture.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a dynamic physical model?
   A. A set of algebraic equations without time
   B. A computer program using random numbers
   C. A hardware model whose behaviour changes with time and imitates a real system's time behaviour
   D. A drawing of the system
   Answer: C. It is built from real mechanical, electrical or fluid parts and observed while running, e.g. a ship model in a tank. (Key point: Definition)

2. Which equation of motion describes the spring-mass-damper suspension?
   A. M x'' + K x + D = F(t)
   B. M x'' + D x' + K x = F(t)
   C. M x' + D x'' + K x = F(t)
   D. M x + D x' + K x'' = F(t)
   Answer: B. Newton's second law gives inertia (M x'') plus damping (D x') plus spring (K x) force equal to F(t). (Key point: Suspension equation)

3. What does D^2 = 4 K M indicate?
   A. Under-damped, oscillating forever
   B. Over-damped and slow
   C. No damping
   D. Critically damped: fastest return without oscillation
   Answer: D. D^2 < 4KM is under-damped, D^2 = 4KM is critical damping, and D^2 > 4KM is over-damped. (Key point: Damping condition)

4. In the RLC analogue of the suspension, which mapping is correct?
   A. Mass M - inductance L; damping D - resistance R
   B. Mass M - resistance R; damping D - inductance L
   C. Mass M - capacitance C; spring K - inductance L
   D. Force F - charge q; displacement x - voltage V
   Answer: A. M maps to L, D to R, K to 1/C, displacement x to charge q and force F to voltage V. (Key point: Electrical analogue)

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