University:
Georgia Southern University
Course:
ENGR1121 | Computing in Mechanical Engineering
Academic year:

2024
Views:

343

Pages:

32
Author:

Jalen W.

Paired data • collect paired data in order to understand the characteristics of an object or the behavior of a system • spatial profile (e.g., temperature versus distance) • or a time history (voltage versus time). • cause-and-effect relationships (e.g., force as a function of displacement) or • system output as a function of a changing input parameter (yield of a chemical reaction as a function of temperature). Numerical methods • Real world data is rarely in a ‘usable’ form • Fundamental techniques for operating on real data to recover ‘useful’ information • Interpolation to ‘fill in holes’ in data or estimate missing data points • Curve fitting to remove random ‘noise’ from data samples • Numerical integration to compute ‘area under the curve’ • Numerical differentiation to compute rate of change of data or slope or gradient LINEAR INTERPOLATION Example 7.2 Linear Interpolation in Excel Suppose we are given the two data points (2.0, 5.5) and (2.1, 7.2). Determine the value of y corresponding to x = 2.03 using linear interpolation. Example 7.2 Linear Interpolation in Excel Suppose we are given the two data points (2.0, 5.5) and (2.1, 7.2). Determine the value of y corresponding to x = 2.03 using linear interpolation. Numerical methods • Real world data is rarely in a ‘usable’ form • Fundamental techniques for operating on real data to recover ‘useful’ information • Interpolation to ‘fill in holes’ in data or estimate missing data points • Curve fitting to remove random ‘noise’ from data samples • Numerical integration to compute ‘area under the curve’ • Numerical differentiation to compute rate of change of data or slope or gradient THE METHOD OF LEAST SQUARES • to fit a curve (including a straight line, as a special case) through an aggregate of data. • It is widely used when working with measured data, where data scatter is an important consideration. • the overall strategy in fitting a curve to a set of data points is to determine an appropriate set of values for the coefficients in the equation of the chosen curve. FITTING A STRAIGHT LINE TO A SET OF DATA determine the values of a and b in the equation y = ax + b in such a manner that the sum of the squares of the errors will be minimized. Assessing a Curve Fit • sum of the squares of the errors (SSE) • Another indication of the quality of the curve fit is the so-called r-squared value • where SST represents the sum of the squares of the deviations about the mean • The r-squared value varies between 0 and 1. • Note that r2 will equal 1 when SSE equals zero. • Hence, an r2 value close to 1 (which means that the sum of the square errors is small) generally indicates a good fit. Example 7.5 Fitting a Straight Line to a Set of Data in Excel • An engineer measured the force exerted by a spring as a function of its displacement from the equilibrium position. Here are the data: The data are plotted as an x-y graph against a background of horizontal and vertical gridlines. Note that the individual data points are not interconnected. after right-clicking on one of the data points. We then select Add Trendline from the resulting menu. This will result in the Format Trendline dialog box shown As an alternative, we could have simply clicked anywhere within the graph to make it active, selected the Ribbon’s Chart Tools/Design tab, and then selected Add Chart Element within the Chart Layouts group. dialog box shown determination of the trendline equation and the r2 value is entirely automatic. Example 7.6 Use of the Regression Feature in Excel using the Regression feature found in the Excel Analysis Toolpak select Data Analysis/Regression from the Analyze group within the Ribbon’s Data tab The coefficients in the regression equation are given in cells E19 and E20, under the heading Coefficients. converted to an x-y graph, (right-click on any of the bars representing the predicted data points and select Change Series Chart Type). The graph can then be expanded (by dragging a corner) so that it can be read more easily. FITTING OTHER FUNCTIONS TO A SET OF DATA • Exponential Functions • let u = ln y and c = ln a, Example 7.8 Fitting an Exponential Function to a Set of Data in Excel The transient behavior of a capacitor has been studied by measuring the voltage drop across the device as a function of time. FITTING OTHER FUNCTIONS TO A SET OF DATA • Logarithmic Functions • Power Functions Example 7.9 Fitting a Power Function to a Set of Data • A chemical engineer is studying the rate at which a reactant is consumed in a chemical reaction involving the manufacture of a polymer. The following data have been obtained, showing the reaction rate (in moles per second) as a function of the concentration of the reactant (moles per cubic foot). Polynomials - method of least squares Example 7.10 Fitting a Polynomial to a Set of Data • The following table presents the time for a high-performance sports car to reach various speeds. • The top speed is considered to be the independent variable in this application, and time is the dependent variable. • Fit an equation to the data, thus obtaining an accurate mathematical relationship between acceleration time and top speed. a straight-line curve fit is not satisfactory SELECTING THE BEST FUNCTION FOR A GIVEN DATA SET 1. 2. 3. 4. 5. Select an equation based upon the underlying theory governing the process used to collect the data. Try to plot the data as a straight line. If this is successful, it will suggest the function that should be used to represent the data. If a straight-line relationship cannot be obtained, try fitting different types of curves to the data. Use visual assessment to recognize a good fit, aided by the resulting values for the sum of the squares of the errors (SSE) and the r2 parameter. If a satisfactory curve fit cannot be obtained using the first two guidelines, try plotting the data differently (for example, try plotting y versus 1/x, 1/y versus x, etc.). In some cases, a better fit may be obtained by scaling the data so that the magnitude of the x-values is more or less the same as the y-values.

Fitting Equations to Data

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