Teaching Mathematics Using Theoretical Model

Abstract

Using the deflection U of a system, the stiffness of truss is interpolated from the experimental data. The linear interpolation is used to estimate the stiffness of a truss using MS Excel’s Forecast function and regression line via VBA’s user-defined function LinearInterpolationK. The change of force, velocity, stiffness and deflection is solved in time using Euler’s Integration Method via VBA worksheet and via user-defined function EulerMethod (Roy, 2010). The trigonometric functions and transformation matrices are used to show the motion of a wheel larger or smaller (scaling in 2D), rotated (rotation in 2D) and shifted (translation in 2D). As it is shown, this model is good not only for teaching mathematics, but also for informatics (the use of Excel’s Worksheet functions, VBA and VBA “Module” feature)... The purpose of this article is to show how mathematics is applied in physics and how this might help students during their particular modules that include algebra or geometry. The physical wheel model was created by Marius Andrikaitis using Microsoft (MS) Excel and Visual Basic for Applications (VBA). The user-defined functions showing parts of model’s mechanism were created by Aurelija Sakalauskaite using MS Excel’s “Module” feature in VBA. The physical wheel model shows how given variables, such as mass, velocity, stiffness of tire and truss, change in time, height, damping coefficient of a tire, lift force, and change in time of a graph, students can see physical object (in this case, wheel) in motion and check how these variables change in time. The wheel model is based on differential calculus (solving second order nonhomogeneous differential equation using Euler’s method), linear algebra (solving two linear equations using finite element analysis (FEA)), geometry (using trigonometric functions and transformation matrices: translation, rotation and scaling). The idea of solving physical problem using common MS program Excel and VBA was inspired by engineer George Lungu, who creates physical models using MS Excel and shares his ideas on his website “Excel Unusual” (Lungu, 2011). One of the main methods that are used in the model is FEA, in particular dividing springs into elements (Widas, 1997). FEA is based on two-spring model, which has 3 degrees of freedom, i.e., three nodes are required to estimate the deflection U of a system (Hutton, 2004). FEA uses inverse matrix and matrix multiplication in solving system of two linear equations.