similarity mechanics.ppt
TRANSCRIPT
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SIMILARITY MECHANICS
Dr.V.G. Idichandy
Professor
Dept of Ocean Engineering,IIT Madras
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Introduction
Model mechanics is concerned with theextent to which inferences drawn fromobservations of physical phenomena on a
mechanical system are quantitativelyapplicable on an analytically similarsystem of different scale.
Similitude here relates to geometric,kinamatic and dynamic.
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Introduction
In cases where the actual mechanical laws
(governing equations) for the phenomenon
under consideration are known, these may
be applied and transferred to model and
prototype provided that the assumption on
which the equations are derived are validfor both the systems.
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Example-Clamped Plate
The similarity condition for a model test on a
flexurally loaded elastic plate will be deduced
from the differential equation for such a plate.
The differential equation for prototype is given
by,
3
2
4
4
22
4
4
4
)1(122
Eh
P
y
w
yx
w
x
w
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Example-Clamped Plate
The same equation for the model
becomes,
here the prime denotes the quantities for
the model.
'h'E
'P)1(12
y
'w
yx
'w2
x
'w 2'4'
4
2'2'
4
4'
4
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Example-Clamped Plate
The necessary conditions for the existence
of scale factors whereby the differential
equation of model and prototype can be
transformed into each other can be arrived
at as follows.
yy
xx
ww
l
l
w
'
'
'
'
'
'
'
hh
EE
pp
h
E
p
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Example-Clamped Plate
If these constants exist, then
To transform the model equation into
prototype equation, it is necessary tosatisfy the condition
33
22
4
4
22
4
4
4
4 )1(12
2
hE
P
y
w
yx
w
x
w
hE
p
l
w
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Example-Clamped Plate
The first equation gives the most general relation to be
satisfied by the scale for deflection, plan dimensions,plate thickness, modulus of elasticity and load.
1
14
3
pl
Ehw
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Example-Clamped Plate
If a complete geometric similarity is presentthen,
In other words if = , the deformations in theplate are dependent upon the similarity factor, ifload per unit area is proportional to the ratios ofE.
hlw
E
E
P
P
PE
''
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There are many phenomena which are ofpractical importance but cannot berepresented by a set of equation.
It is for these types of problems model testare some times the only means of getting
precise information. For such casesscaling laws can be done only bydimensional analysis
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Dimensions and dimensional
homogeneity
Dimensional analysis starts from the assumptionthat a generally valid physical law must bedimensionally homogeneous, i.e. the dimensionmust be equal for all the terms of a sum.
Qualitatively, a physical phenomena can beexpressed in certain fundamental measures of
nature, called dimensions. Our problem beingmechanical the dimensions are length (L), force(F) and time (T).
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Quantitatively, the phenomena has both
number and a standard for comparison.
However, regardless of the system, the
governing equations must be valid or they
are dimensionally homogeneous. When it
is so, any equation of the form F(x1, x2..xn) = 0
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The expression can also be expressed in the form
where terms are dimensionless products of n
physical variables (x1, x2..xn) and
m= (nr) where r is the number of fundamental
dimensions that are involved in n physicalvariables.
0).......,(G m21
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This has two very important implications.
The form of physical occurrence may be partiallyreduced by proper consideration of dimensionsof n quantities identified to influence theoccurrence.
Physical quantities of prototype and reducedscale model will have identical functionals G.
Similitude requirements for modeling result fromforcing the terms (1, 2m) to be equal inmodel and prototypes.
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Dimensional analysis
Dimensional analysis is of substantial
benefit in any investigation of physical
behaviour because it permits the
experimenter to combine variables intoconvenient grouping (terms) with a
subsequent reduction in the unknowns.
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Buckingham Pi theorem
The theorem states that any dimensionally
homogeneous equation involving certain
physical quantities can be reduced to an
equivalent equation involving a completeset of dimensionless product. the number
of such independent terms will be
m = n - r where n is the number of physicalquantities and r the number of dimensions.
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F(X1, X2Xn) = 0
Can also be expressed in the form
0).......,(G m21
2654321 ElP,,,
RP,1,
E
etc
E
,
ER
P,
P
l2
1 E P R
L 1 -2 0 0 0 -2 1 0
T 0 1 0 1 1 1 0 0
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numberynoldsRevl1
tCoefficienessurePr
v
P22
numberFroudelg
v2
3
numberMachc
v4
numberWeberlv
2
5
1 P v g c
L 1 -1 1 -3 1 1 1 0
T 0 -2 -1 0 -2 -1 -1 -2
M 0 1 0 1 0 0 0 1
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2m
2
cr
1 a
ha2
w
m3
a.a.
h
'h'a 1
h a m w cr
M 0 0 1 1 1 0
L 1 1 -3 -3 -1 0T 0 0 0 0 -2 -1
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cr
cr
1
''
hg m
cr4
cr
cr
m
m
cr
cr
m
m '
''h
h.g'gor
'
hg
'g'h
1 T 1 E V
M 0 0 0 1 1 1 1 0 0
L 1 1 0 -1 -3 -3 -1 0 1
T 0 0 0 -2 0 0 -2 -1 -1
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E
'E
v'v
E
'E'
1
The terms are
E
v,v
l,E
,,,L
2
654
L
T321