Application of interpolation and finite difference

Application of Interpolation and
Finite Difference
By : Parth Luhar(43)
N Chaitanya Sai (53)
Manthan Chavda (18)
Parth Patel (72)
Krutarth Patel(75)

Interpolation
• Interpolation is a technique of finding new
data points within the range of known data
points.
• Y2 = (x2-x1) (y3-y1) + y1
(x3-x1)

Application of interpolation
• A spring is an elastic object used to
store mechanical energy.
• The deflection is plotted on the x-
axis and the corresponding load in
newton on y-axis. Often we have
to find the values between the two
sets of values (load vs. deflection).
Hence interpolation is the
technique used to find the
unknown values.

The figure above shows the spring deflection at values of 7.5 and 14.33mm
and the corresponding load. One is the graph for minimum load values shown
in blue (157.31 and 79.05). The other graph shown in red is for the maximum
force values (178.69 and 90.95).

APPLICATION OF THE POLYNOMIAL
INTERPOLATION METHOD FOR
DETERMINING PERFORMANCE
CHARACTERISTICS OF A DIESEL ENGINE

The study was conducted on a typical engine
dynamometer, the diagram of which was
shown in Fig. 1.

Application of interpolation and finite difference

Finite difference
• Some operators should be known before going
deep in application of finite difference.
• Forward difference
• Backward difference
• Central difference
• Shift operator

Shift Operator (E)
When this operator is applied on f(xi) we’ll get
like following ,
Ef(xi) = f(xi + h) = f(xi+1)
Ef(x0) = f(x0 + h) = f(x1), Ef(x1) = f(x1 + h) = f(x2)
• Commonly,
Ek f(xi) = f(xi + kh) = f(xi+k)
• For k=1/2

Methods For Interpolation
• Newton’s Forward Interpolation formula
• Newton’s Backward Interpolation formula

Newton’s forward
• Suppose h be the length of the given table ,
then S be the term which is mathematically
described as ,
s = [(x – x0)/h] .
Final formula given by,
f(x)

Newton’s backward
• Suppose h be the length of the given table ,
then S be the term which is mathematically
described as ,
x – xn = sh
• Final formula given by,
f(xn + sh) = f(xn) + s∇f(xn) +

Application in my field
• Cantilever Thin Beam with Point Load at Free
End

Contd.
EI
Integrate with respect to, we get
Integrate again and we get

Contd.
• A and B are constants of integration and must
be found from the boundary conditions.
• These are at (no deflection)
and at (gradient horizontal)
• Hence, A= & B=

Contd.
• Now numerical analysis ,calculating on given
data that A cantilever thin beam is 4m long
and has a point load of 5KN at the free end.
The flexural stiffness is 53.3MN2

Contd.
• We’ll get a table between slope and length of
this cantilever beam,
From this table we can calculate slope
anywhere on beam using interpolation.
If we required to calculate slope at 1.8 m from
free end we can use following procedure,

Contd.
X Y
0 1.688*10^-3 -1.88*10^-4 -3.74*10^-4
2 1.5*10^-3 -5.62*10^-4 -3.76*10^-4 -.01*10^-4
4 9.38*10^-4 -9.38*10^-4
6 00

Contd.
S=(1.8-2)/2=-0.1
Doing interpolation using newton’s forward
formula
Y=1.5*10^-3+(-0.1*9.38*10^-4)=1.4062*10^-3

Contd.
International Journal of Mechanical
Engineering and Technology (IJMET),
ISSN 0976 –
6340(Print), ISSN 0976 –
6359(Online) Volume 4, Issue 2,
March - April (2013) © IAEME

Conclusion
• We can calculate slope & deflection in any
type of beam ( cantilever ,UDL ,UVL ,Point
loaded) using this interpolation formulas if we
have already estimated slope and deflection
of beam at finite points.

Refrences
[1] Zhang, G.Y., 2010, “A Thin Beam Formulation Based on Interpolation Method”,
International Journal of numerical methods in engineering, volume 85, pp. 7-35.
[2] Wang, Hu, Guang, Li Yao, 2007, “Successively Point Interpolation for One
Dimensional Formulation”, Engineering Analysis with Boundary Elements, volume
31, pp. 122-143.
[3] Ballarini, Roberto, S, 2003 “Euler-Bernoulli Beam Theory”, Mechanical Engineering
Magazine Online.
[4] Liu, Wing Kam, 2010, “Meshless Method for Linear One-Dimensional Interpolation
Method”, International Journal of Computer Methods in Applied Mechanics and
Engineering, volume 152, pp. 55-71.
[5] Park, S.K. and Gao, X.L., 2007, “Bernoulli-Euler Beam Theory Model Based on a
Modified Coupled Stress Theory”, International of Journal of Micro-mechanics and
Micro- engineering, volume 19, pp. 12-67.
[6] Paul, Bourke, 2010, “Interpolation Method”, International Journal of Numerical
Methods in Engineering, volume 88, pp. 45-78.
[7] Launder, B.E. and Spading D.B., 2010, “The Numerical Computation of Thin
Beams”, International Journal of Computer Methods in Applied Mechanics and
Engineering, volume 3, pp. 296-289.
[8] Ballarini and Roberto, 2009, “Euler-Bernoulli Beam Theory Numerical Study of Thin
Beams”, International of Computer in Applied Mechanics and Engineering, volume
178, pp. 323-341.
[9] Thomson, J.F., Warsi Z.U.A. and Mastin C.W., 1982 “Boundary Fitted Co-ordinate
system for Numerical Solution of Partial Differential Equations”, Journal of
Computational Physics, volume 47, pp. 1-108.
[10] Gilat, Amos, January 2003, “MATLAB An Introduction with Application,
Publication- John Wiley and Sons.
[11] Hashin, Z and Shtrikman, S., 1963 “A Variation Approach to the Theory of Elastic
Behaviour of Multiphase Materials”, Journal of Mechanics and Physics of Solids,
volume 11, pp. 127-140.
[12] Liu, G.R. and Gu, Y.T., 2001, “A Point Interpolation Method for One-Dimensional
Solids”, International Journal of Numerical Methods Engineering, pp. 1081-1106.
[13] Katsikade, J.T. and Tsiatas, G.C., 2001, “Large Deflection Analysis of Beams with
Variable Stiffness”, International Journal of Numerical Methods Engineering, volume
33, pp. 172-177.

Application of interpolation and finite difference

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