Dimensional Analysis

Introduction

  • Curiosity, NASA's Mars Rover, landed on Mars on August 5, 2012.

    • Entered atmosphere at 20,000 km/h, slowed to 1,600 km/h via drag.

    • Parachute opened, reducing speed to 320 km/h (90 m/s).

    • Final deceleration accomplished by rockets, successfully landing on the Martian surface.

  • Landing sequence strategies devised through rigorous Earth-based experiments.

  • Dimensional analysis is the problem-solving method employed to ensure designs function correctly on Mars.

Dimensional Analysis

  • Problem-solving skill critical for design and optimization of systems.

  • Presented by Ken Kamrin, mechanical engineering professor at MIT.

  • Importance of understanding unit analysis and independent vs. dependent variables.

  • Objective: Use dimensional analysis to estimate the parachute canopy size to achieve 90 m/s terminal velocity for the Rover.

Understanding Dimensions and Units

  • Distinction between dimensions and units:

    • Units refer to specific measurements (e.g., grams, kilograms).

    • Dimensions represent the nature of a physical quantity (e.g., mass).

  • Five fundamental dimensions:

    1. Length (L)

    2. Mass (M)

    3. Time (T)

    4. Temperature

    5. Charge

  • Derived dimensions through combinations and powers of fundamental ones.

  • For example, velocity has dimensions of L/T:

    • Energy: M L^2 T^-2.

Designing the Parachute

  • Goal: Design parachute to slow Rover to precisely 90 m/s.

  • Factors affecting terminal velocity include:

    • Parachute canopy diameter (independent variable)

    • Mass of the Rover

    • Acceleration due to gravity

    • Density and viscosity of the atmosphere

  • Viscosity assumed negligible due to similarities in atmospheres.

Key Variables

  • Dependent Variable:

    • Terminal velocity (the desired speed for safe landing).

  • Independent Variables:

    1. Canopy diameter

    2. Rover mass (ignoring parachute mass)

    3. Gravitational acceleration

    4. Atmospheric density

  • Surface area not treated as an independent variable due to dependence on diameter.

Experiments and Relationships

  • Express terminal velocity as a function of independent variables via experiments.

  • Ensure all terms in equations share the same dimensions, restricting possible functional forms.

  • Dimensional analysis will yield dimensionless expressions leading to a general functional relationship.

Steps in Dimensional Analysis

  1. Identify Variables: Capture dependent and independent variables.

  2. Distill Dimensions: Express dimensions of dependent and independent variables.

    • Example: Distilling gravity leads to dimensions of L/T².

  3. Create Dimensionless Quantities:

    • Terminal velocity (v): L/T → dimensionless by multiplying by time and dividing by length.

  4. Establish Relationships:

    • Combine dimensionless variables to form a function of terminal velocity.

  5. Rearrange to Formula:

    • Derive a usable formula for terminal velocity with correct dimensions.

  6. Experimentation:

    • Test hypotheses on Earth and adapt findings to Martian conditions.

Practical Application of Analysis

  • A specific function can be identified that matches the computed dimensions.

  • Explore relationships using different choices of fundamental dimensions, while reaching consistent results.

  • Adapt findings from Earth experiments for rover landing on Mars, using known parameters for Martian gravity and atmosphere.

Conclusion

  • Dimensional analysis serves as a robust tool for engineers to transfer knowledge from one context to another, particularly when direct experimentation is impractical.

  • This methodology allows for more efficient design processes and can illuminate solutions to challenging engineering problems.