Table 2.
A Dimensional Matrix Showing the Relationship Between Each Continuum Variable and Its Fundamental Dimensions (L for Length, T for Time, and M for Mass)a
| ρ | t | x | y | z | ux | uy | uz | gx | gy | gz | A | σxx | σxy | σxz | σyy | σyz | σzz | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L | –3 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | n | –1 | –1 | –1 | –1 | –1 | –1 |
| T | 0 | 1 | 0 | 0 | 0 | –1 | –1 | –1 | –2 | –2 | –2 | 2n – 1 | –2 | –2 | –2 | –2 | –2 | –2 |
| M | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | –n | 1 | 1 | 1 | 1 | 1 | 1 |
From the Buckingham Pi Theorem, the rank of the dimensional matrix gives the minimum number of variables necessary to reproduce the dimensional structure of all other variables. For this matrix, the rank is three. By inspection, the three columns associated with ρ, t, and x are independent and contain all three dimensions. These three columns form a set of basis vectors that can be used to construct the dimensions of each of the other variables. Other basis vectors are obvious from the table (e.g., A, ρ, and z), but a different choice gives only a recombination of the dimensionless parameters predicted by the original choice. For details about dimensional matrices, we recommend Welty et al. [1984].