Nonlinear Curve Fitting Techniques in Engineering

Utilizing nonlinear curve fitting techniques is crucial in engineering to analyze data relationships that are not linear. This involves transforming nonlinear equations into linear form for regression analysis, as demonstrated in examples and methods such as polynomial interpolation and exponential curve fitting. Dr. Mohamed El-Shazly, an expert in Mechanical Design and Tribology, guides students in the exploration of numerical methods for curve fitting in the Mechanical Engineering Department.

• Nonlinear Curve Fitting
• Engineering
• Numerical Methods
• Mechanical Engineering
• Dr. Mohamed El-Shazly

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1. Faculty of Engineering Mechanical Engineering Department MATH 2140 Numerical Methods Instructor: Dr. Mohamed El-Shazly Associate Prof. of Mechanical Design and Tribology melshazly@ksu.edu.sa Office: F072 1

2. Curve-Fitting Polynomial Interpolation 2

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5. CURVE FITTING WITH NONLINEAR EQUATION BY WRITING THE EQUATION IN A LINEAR FORM Many situations in science and engineering show that the relationship between the quantities that are being considered is not linear. For example, Fig. 6-8 shows a plot of data points that were measured in an experiment with an RC circuit. In this experiment, the voltage across the resistor is measured as a function of time, starting when the switch is closed. 5

6. Examples of nonlinear functions used for curve fitting in the present section are: 6

7. Writing a nonlinear equation in linear form In order to be able to use linear regression, the form of a nonlinear equation of two variables is changed such that the new form is linear with terms that contain the original variables. For example, the power function can be put into linear form by taking the natural logarithm (ln) of both sides: 7

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10. EXAMPLE 1 Find the curve of best fit of the type : to the following data by the method of least squares: 10

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12. EXAMPLE 2 For the data given below, find the equation to the best fitting exponential curve of the form: And if you need to convert between them: log10(x) = ln(x) / ln(10) ln(x) = log10(x) / log10(e) 12

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14. EXAMPLE 3 Given: xi 1 2 3 yi 2.4 5 9 bx f(x) = function a Find that best fits the data. ae 14

15. Linearization Method bx = function a Find that best fits the data. f(x) ae )) = = + Define ( ) ln( ( ln( ) g x f x a b x = = + Define = ln( and ) ln( = ) z y a bx i i i Let ln( ) ln( ) a z y i i 15

16. Evaluating Sums and Solving xi yi zi=ln(yi) xi2 xizi 1 2.4 0.875469 1 0.875469 2 5 1.609438 4 3.218876 3 9 2.197225 9 6.591674 =6 =4.68213 =14 =10.6860 Equations : = = ln( ), a a e 23897 . 0 = 26994 . 1 = a e Solving = Equations : 66087 . 0 bx x = 26994 . 1 = ( ) f x ae e = 23897 . 0 , 66087 . 0 b 16

17. Given the data: x 0 1 2 3 4 y 1.5 2.5 3.5 5 7.5 = e y a Find the exponential function fit to this data. bx 17

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