Interpolation

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Interpolation

• Topic Spline Interpolation Method
• Major Mechanical

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What is Interpolation ?

Given (x0,y0), (x1,y1), (xn,yn), find the
value of y at a value of x that is not given.
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Interpolants

• Polynomials are the most common choice of
  interpolants because they are easy to
• Evaluate
• Differentiate, and
• Integrate.

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Why Splines ?
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Why Splines ?
Figure Higher order polynomial interpolation is
a bad idea
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Linear Interpolation
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Linear Interpolation (contd)
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Example

• A trunnion is cooled 80F to - 108F. Given
  below is the table of the coefficient of thermal
  expansion vs. temperature. Determine the value of
  the coefficient of thermal expansion at T-14F
  using linear spline interpolation.

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Linear Interpolation




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Quadratic Interpolation
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Quadratic Interpolation (contd)
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Quadratic Splines (contd)
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Quadratic Splines (contd)
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Quadratic Splines (contd)
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Example

• A trunnion is cooled 80F to - 108F. Given
  below is the table of the coefficient of thermal
  expansion vs. temperature. Determine the value of
  the coefficient of thermal expansion at T-14F
  using quadratic spline interpolation.

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Solution




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Solution (contd)
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Solution (contd)
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Solution (contd)
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Solution (contd)
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Solution (contd)
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Reduction in Diameter
The actual reduction in diameter is given by
where Tr room temperature (F) Tf
temperature of cooling medium (F) Since Tr 80
F and Tr -108 F, Find out the percentage
difference in the reduction in the diameter by
the above integral formula and the result using
the thermal expansion coefficient from the cubic
interpolation.
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Reduction in Diameter
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Reduction in diameter
Taking the average coefficient of thermal
expansion over this interval, given by
The absolute relative approximate error
obtained between the results from the 2nd methods
is