10.3 The physics of springs  (Page 4/11)

Ideally, we think we are solving the equation $A^{T}Ek=f$ , where $E\in {\mathbb{R}}^{16\times 16}$ has the elongation of each spring along the diagonal. In reality, we are actually attempting to solve the relation

where α , ϵ , κ , and γ are error due to either the measurements or the model. We would like to use statistical inference to find a solution.

Applications

Our experiment is designed to test the elastic properties of a very simple network by stretching it in two dimensions. This is an instance of the biaxial test, which is used to study the properties of polymers, plastics, and tissue by material scientists and physicians. Our setup simulatesa biopsy that seeks to identify a flaw in an otherwise homogeneous material.

Notes: our data sets, measuring spring constants, and error

We ran two large sets of experiments. Data Set A consists of 104 experiments on Network g (see "Conditioning" for a list of all networks). All the even experiments have (theoretically) identical forces; all the odd experiments have another set of (theoretically) identical forces. We are confident of the magnitude of the forces; we are less confident of the alignment. These 104 experiments are actually just two experiments, each repeated 52 times. Data Set B consists of 120 unique experiments on Network a. No set of forces was repeated. These two data sets will allow us to explore which sets of experiments produce the best results, a most important question for applications. Data Set A will allow us to approach the problem statistically.

In order to be able to say which set of experiments or computational method gives us the “best” results, we need to know the actual values of the spring constants. We would like each spring of a particular class (horizontal/vertical, diagonal, stiff) to have the same k , but because these springs were purchased at our local hardware store, we have no guarantee of that and much measure each spring individually.

To measure k for a particular spring, we attach the spring to one of the fixed nodes, with a penny and string on the other end. We begin with only the 50g hook on the string and use the webcam and MATLAB to take the position. Next, we add 10g to the hook at a time, taking the position after adding each mass, until 100g are on the hook. This gives us 11 data points to work with. We plot these, then find the line of best fit through these points using three different methods: MATLAB's polyfit function; the least squares approach, MATLAB's backslash, which forms the normal equations $k=e^{T}f/e^{t}e$ ; and the total least squares approach, described in "Total Least Squares" . The results from each of these methods is recorded in [link] .