Newton’s Divided Difference Interpolation-More Examples ...
Chapter 05.03
Newton’s Divided Difference Interpolation –
More Examples
Industrial Engineering
Example 1
The geometry of a cam is given in Figure 1. A curve needs to be fit through the seven points given in Table 1 to fabricate the cam.
|[pic] |
|Figure 1 Schematic of cam profile. |
| Table 1 Geometry of the cam. |
|Point |
|[pic][pic] |
|[pic][pic] |
| |
|1 |
|2.20 |
|0.00 |
| |
|2 |
|1.28 |
|0.88 |
| |
|3 |
|0.66 |
|1.14 |
| |
|4 |
|0.00 |
|1.20 |
| |
|5 |
|–0.60 |
|1.04 |
| |
|6 |
|–1.04 |
|0.60 |
| |
|7 |
|–1.20 |
|0.00 |
| |
If the cam follows a straight line profile from [pic] to [pic], what is the value of [pic] at [pic] using Newton’s divided difference method of interpolation and a first order polynomial.
Solution
For linear interpolation, the value of [pic] is given by
[pic]
Since we want to find the value of [pic] at [pic], using the two points [pic] and [pic], then
[pic] [pic]
[pic] [pic]
gives
[pic]
[pic]
[pic]
[pic]
[pic]
Hence
[pic]
[pic] [pic]
At [pic]
[pic]
[pic]
If we expand
[pic] [pic]
we get
[pic] [pic]
This is the same expression that was obtained with the direct method.
Example 2
The geometry of a cam is given in Figure 2. A curve needs to be fit through the seven points given in Table 2 to fabricate the cam.
|[pic] |
|Figure 2 Schematic of cam profile. |
| Table 2 Geometry of the cam. |
|Point |
|[pic][pic] |
|[pic][pic] |
| |
|1 |
|2.20 |
|0.00 |
| |
|2 |
|1.28 |
|0.88 |
| |
|3 |
|0.66 |
|1.14 |
| |
|4 |
|0.00 |
|1.20 |
| |
|5 |
|–0.60 |
|1.04 |
| |
|6 |
|–1.04 |
|0.60 |
| |
|7 |
|–1.20 |
|0.00 |
| |
If the cam follows a quadratic profile from [pic] to [pic] to [pic], what is the value of [pic] at [pic] using Newton’s divided difference method of interpolation and a second order polynomial. Find the absolute relative approximate error for the second order polynomial approximation.
Solution
For quadratic interpolation, the value of [pic] is chosen as
[pic]
Since we want to find the value of [pic] at [pic] using the three points [pic], [pic] and [pic], then
[pic] [pic]
[pic] [pic]
[pic] [pic]
gives
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Hence
[pic]
[pic] [pic]
At [pic]
[pic]
[pic]
The absolute relative approximate error [pic] obtained between the results from the first and second order polynomial is
[pic]
[pic]
If we expand
[pic] [pic]
we get
[pic] [pic]
This is the same expression that was obtained with the direct method.
Example 3
The geometry of a cam is given in Figure 3. A curve needs to be fit through the seven points given in Table 3 to fabricate the cam.
|[pic] |
| |
|Figure 3 Schematic of cam profile. |
| |
| |
|Table 3 Geometry of the cam. |
|Point |
|[pic][pic] |
|[pic][pic] |
| |
|1 |
|2.20 |
|0.00 |
| |
|2 |
|1.28 |
|0.88 |
| |
|3 |
|0.66 |
|1.14 |
| |
|4 |
|0.00 |
|1.20 |
| |
|5 |
|–0.60 |
|1.04 |
| |
|6 |
|–1.04 |
|0.60 |
| |
|7 |
|–1.20 |
|0.00 |
| |
Find the cam profile using all seven points in Table 3, Newton’s divided difference method of interpolation and a sixth order polynomial.
Solution
For 6th order interpolation, the value of [pic] is given by
[pic]
Using the seven points,
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
gives
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic] [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Hence
[pic]
[pic]
Expanding this formula, we get
[pic]
This is the same expression that was obtained with the direct method.
|[pic] |
|Figure 4 Plot of the cam profile as defined by a 6th order interpolating polynomial (using Newton’s divided difference method of |
|interpolation). |
|INTERPOLATION | |
|Topic |Newton’s Divided Difference Interpolation |
|Summary |Examples of Newton’s divided difference interpolation. |
|Major |Industrial Engineering |
|Authors |Autar Kaw |
|Date |August 12, 2009 |
|Web Site | |
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