11DIMENSIONALANALYSIS.ppsx

7/28/2019 11DIMENSIONALANALYSIS.ppsx

1/11

DIMENSIONAL ANALYSISEngineering equations:

DIMENSIONALLY HOMOGENOUS:

dimensions in equation balance; constants dimensionless;

rational derivation from 1st principles.

EMPIRICAL:

requires constants with the right units;

empirical experimentation (effect vs cause plot).


2/11

DIMENSIONAL ANALYSISDimensional Analysis only valid when equations are

DIMENSIONALLY HOMOGENOUS.

Units broken into primary units of measurement: Length L;

MassM; and

Time T.


3/11

DIMENSIONAL ANALYSISPROBLEM 1

Is the following well-known kinematic equation

dimensionally homogenous?Where

a acceleration

d distance

Vo velocity

t time

t

2V

t

2da 0

2

m/s2

m

m/s

s

(Answer: Yes)

222s

2m

s

2m

s

m


4/11

DIMENSIONAL ANALYSISPROBLEM 2What should the units of A be to render the following

equation dimensionally homogenous?

Where

E Bulk modulus

Poissons ratiod, y, h distances

R ratio of distances

F force

MPaMN/m2

-m

-

N

(Answer: A L)

3

22 21

4

AA

y

hyhRd

Ey

F


5/11

DIMENSIONAL ANALYSISDimensional Analysis: technique to establish

relationship amongst variables by looking at units;

Powerful tool for analysis and teaching especiallywith multiple variables;

Choose variables with bearing on math. relationship;

Relate math. through exponential relationship

(unknown exponents) solve exponents


6/11

DIMENSIONAL ANALYSISGeneral Approach:

Formulate an unknown relationship in general terms;

Balance each of the fundamental units;

Solve the simultaneous linear equations; and

Express the resulting relationship.


7/11

DIMENSIONAL ANALYSISFinding dimensionless groups from first principles:

Formulate an unknown relationship in general terms;

Balance each of the fundamental units;

Solve the simultaneous linear equations in terms of

remaining unknown exponents; and

Express the resulting relationship.


8/11

DIMENSIONAL ANALYSISEXAMPLE Force derived from a propellerThe thrust F developed by a propeller spinning in a

liquid depends on five variablesthe density andviscosity of the liquid, the diameter and rotational speed

of the propeller, and the velocity at which the propeller

is moving through the liquid. From first principles,

show that:

v

nd,

dv

.v.dF 22


9/11

DIMENSIONAL ANALYSISEXAMPLE Force derived from a propeller

thrust F M.L.T-2;density M.L-3;

viscosity M.L-1.T-1,

diameter d L;

rotational speed n T-1;

velocity v L.T-1.

The general relationship is:

v

nd,

dv

.v.dF 22

edcba.n.

.

.vK.dF


10/11


edcba.n...vK.dF

eddd3ccbba2.T.T.L.M.L.M.T.LLM.L.T

d3cba1:L

edb2:T

dc1:M

c = 1d

b = 2de

a = 2d + e


11/11


edd1ed2ed2.n...vK.dF

eeedddd22 .n.vd....vd...vK.dF

v

nd,

dv

.v.dF 22

11DIMENSIONALANALYSIS.ppsx

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