# Dimensional Analysis

## Work with units to perform dimensioned calculations

Dimensional analysis lets you understand the relationship between different physical quantities in equations. A dimension—for example, length—is a measure of a physical quantity without a number. A unit—for example, meter, inch, or mileassigns a number or measurement to that dimension.

With dimensional analysis you can:

• Work with standard units and unit systems. Unit systems comprise of a set of base units and derived units. The most common unit systems are SI, US, and CGS. For example, SI base units include mass (kg), time (s), length (m), electric current (ampere), and temperature (kelvin). Derived units such as density ($$kg/m^{3}$$) or force ($$kg\ m/s^{2}$$) can be expressed as a combination of the base units. You can perform calculations with base units or create new units and unit systems.
• Convert between units. In dimensional analysis, a ratio that converts one unit of measure into another without changing the quantity is called a unit conversion factor. For example, the unit conversion factor between inch and cm is 127/50. You can compute unit conversion factors between Celsius, Fahrenheit, and Kelvin and switch between absolute temperature and temperature difference.
• Check for unit consistency. You can use dimensional analysis to check that the equations you derive accurately represent the physics by checking the consistency of units. Equations with valid physical meaning will have the same units on the left- and right-hand sides.
• Perform scale analysis. You can use dimensional analysis to understand and maintain accuracy in numerical methods—for example, finite difference approximations with numerators and denominators that differ by orders of magnitudes.
• Find dimensionless characteristic constants. You can nondimensionalize equations and determine unitless characteristic constants that can be useful in describing the behavior of a system. For example, you can find the damping ratio ζ (zeta) of a mass spring system and use this to describe whether the spring system is underdamped (ζ < 1), overdamped (ζ > 1), or critically damped (ζ = 1).

## Units in MATLAB

Symbolic Math Toolbox™ supports over 2000 units of measurement and, using the MATLAB Live Editor, displays units in blue type, making them easy to identify. For more information on dimensional analysis, see Symbolic Math Toolbox.

## Units in Simulink and Simscape

Simulink® enables you to specify physical units as signal attributes to ensure the consistency of calculations across the various components of your model. For more information on Simulink units, see Unit Specification in Simulink Models. Simscape also supports units in the physical modeling environment. For more information, see physical units.

See also: Computer algebra system, mathematical modeling, analytical solution, Symbolic Math Toolbox, Live Editor