03 Finite Differences

Learning objectives:

1. Can calculate velocity and acceleration from position measurements in time
2. Can account for the errors in the position measurements

Problem: Engineers usually don’t have a perfect function of position over time, if we do, nothing can stop us from conquering the world HAHAHA! Instead we have a a table of position over time, we can use finite differences to calculate velocity and acceleration.

Finite Differences

1. First Order Forward Finite Difference

$$f^{\prime }(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

2. First Order Backward Finite Difference

$$f^{\prime }(x)=\frac{f(x)-f(x-\Delta x)}{\Delta x}$$

3. Second Order Centered Finite Difference

$$f^{\prime }(x)=\frac{f(x+\Delta x)-f(x-\Delta x)}{2\Delta x}$$

As the denominator is larger, Second Order Centered Finite Difference is more accurate! Want to dive more? Learn about Taylor Series!

Treating Data Set

To not make arts in engineering, we should treat the data using moving average:

$${\bar{x}}_i=\frac{x_{i+1}+x_i+x_{i-1}}{3}$$

To gain more accurate data, you can use multiple moving averages or higher order finite differences. However, note that using them means you will loose data points.