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Example: Under
certain conditions, wind blowing past a rectangular speed limit sign
Can
cause the sign to oscillate with frequency, ω. Assume that
ω is a function of the
Sign
width, b, sign height, h, wind velocity, V, air density, ρ, and an
elastic constant, k,
for
the supporting pole. The constant, k,
has the dimensions of FL. Develop a suitable
set
of Pi terms for this application.
|
1. |
Identify
variables governing the problem.
(Given in problem statement.) |
|
2. |
List
dimensions for each variable.
Pick F,L, and T since
dimensions of k are FL
ω
→ 1/T, b → L, h → L, V → L/T, ρ → M/L3 = FT2/L4, k
→ FL
Note: F=Ma
so M = F/a a → L/T2 Therefore
M → FT2/L |
|
3. |
Number
of dimensionless products (Pi terms) = 3
since
number of variables = 6 number of dimensions = 3 |
|
4. |
Select
the dependent variable and the repeating variables.
Dependent
variable = ω, Non-repeating variables: h and k
Repeating
variables: b→L, V→T, ρ→F |
|
5. |
Use
the repeating variables to form dimensionless products by inspection. i.e.
Use b to eliminate the dimension of L, V to
eliminate the dimension of T,
and
ρ to eliminate the dimension of F. |
|
6. |
Form
the pi terms by inspection (appropriate multiplication and division of
variables)
h/b→L/L = 1,
so Pi1 = h/b, ωb/V→(1/T)(L)(T/L) = 1 so Pi2 = ωb/V
k/ρV2b3
→(FL)/[ (FT2/L4)(L/T)2(L3)
= 1 so Pi3
= k/ρV2b3 |
Click
here for this example using the method of exponents. Click here for another example. |