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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.
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1. |
Identify
variables governing the problem.
(Given in problem statement.) |
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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 |
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3. |
Number
of dimensionless products (Pi terms) = 3
since
number of variables = 6 number of dimensions = 3 |
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4. |
Select
the dependent variable and the repeating variables.
Dependent
variable = ω, Non-repeating variables: h and k
Repeating
variables: Use B→L, V→T, ρ→F |
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5. |
Use
V to eliminate T, use b to eliminate
L and use ρ to eliminate
F. Then form the
algebraic
equations involving the exponents as follows:
π1
= ω Ba Vb ρc π2 = h Ba
Vb ρc π3 = k Ba Vb
ρc |
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Click
here to continue with this example. |
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