Dimensional Analysis Tutorial

May 31, 2024

Dimensional Analysis Tutorial

Basic Dimensions

  • Limited set of basic dimensions in physical systems
    • Mass (M): Represented with a capital M
    • Length (L): Distance measurements
    • Time (T): Speeds are lengths per time
    • Temperature: Sometimes considered
  • Other properties like force can be derived
    • Force (F): Mass x Length / Time² (M * L / T²)

Derived Quantities

  • Velocity (V): Length per time (L / T)
  • Acceleration (A): Change in velocity over time (L / T²)

Systems of Units

  • English System: Uses pounds (force), slugs (mass)
  • SI Units (System International)
    • Mass in kilograms (kg)
    • Length in meters (m)
    • Time in seconds (s)
    • Temperature in Kelvin (K)
    • Derived units: e.g., Newton (N) = kg·m/s²

Dimensional Homogeneity

  • All terms in an equation must have the same dimensions
  • Ensures consistency within physical equations
  • Example: Equation of motion

Buckingham Pi Theorem

  • Useful for non-dimensionalizing equations
  • If an equation has k variables, it can be reduced to k - r independent dimensionless products (numbers)
    • r: Minimum number of basic dimensions to describe all variables

Applying Buckingham Pi Theorem

  • For an equation involving variables and parameters:
    • Example: Equation of motion for a bead in a hoop
    • Count variables and basic dimensions
      • Variables (k): 7 (including θ and t)
      • Basic dimensions (r): 3 (Mass, Length, Time)
  • k - r dimensionless numbers
    • 7 - 3 = 4 dimensionless numbers

Finding Dimensionless Numbers

  1. θ: Already dimensionless
  2. γ (Gamma): From ratio of forces: (R * Ω² / G)
  3. τ (Tau): Dimensionless time (t / T)
  4. ε (Epsilon): Derived after normalizing terms
  • Epsilon (ε) = (M² * G * R) / B²

Equation Non-dimensionalization

  • Original and derived equations:
    • Equation becomes in terms of non-dimensional variables
    • Example: Second-order derivatives and damping terms transformed
  • Choices in non-dimensional parameters and characteristic time scale

Famous Examples of Dimensionless Numbers

  • Reynolds Number: Relative importance of inertia to viscosity in fluids
    • Low Reynolds number: Laminar flow
    • Higher Reynolds number: Transition to turbulence
    • Considered a bifurcation parameter