similarity mechanics.ppt

8/14/2019 similarity mechanics.ppt

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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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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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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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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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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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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

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