chapt3 - ME 4905 Advanced Numerical Methods Fourier...

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Unformatted text preview: ME 4905 Advanced Numerical Methods Fourier Approximation Lecturer: T.Y. Ng (PhD) Email: [email protected] Introduction Engineers often deal with systems that oscillate or vibrate. Therefore trigonometric functions play a fundamental role in modeling such problems. Fourier approximation represents a systemic framework for using trigonometric series for this purpose. s A periodic function f ( t ) is one for which where T is a constant called the period that is the smallest value for which this equation holds. ) ( ) ( T t f t f + = Sinusoidal Functions s Any waveform that can be described as a sine or cosine is called sinusoid: The m e a n v a l u e A sets the average height above the abscissa. The a m p l i t u d e C 1 specifies the height of the oscillation. The a n g u l a r f r e q u e n c y ω c h a r a c t e r i z e s h o w o f t e n t h e c y c l e s o c c u r . The phase angle , or p h a s e s h i f t , t parameterizes the extent which the sinusoid is shifted horizontally. ) cos( ) ( 1 θ ω + + = t C A t f Alternative Formulation An alternative model that still requires four parameters but that is cast in the format of a general linear model can be obtained by invoking the trigonometric identity: 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 arctan ) sin( ) cos( where ) sin( ) cos( ) ( )] sin( ) sin( ) cos( ) [cos( ) cos( B A C A B C B C A t B t A A t f t t C t C + =        - =- = = + + =- = + θ θ θ ω ω θ ω θ ω θ ω Least-Squares Fit of a Sinusoid s Sinusoid equation can be thought of as a linear least-squares model s Thus our goal is to determine coefficient values that minimize e t B t A A y + + + = ) sin( ) cos( 1 1 ω ω { } ∑ = + +- = N i i i i r t B t A A y S 1 2 1 1 )] sin( ) cos( [ ω ω Normal Equation           =                     ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ t y t y y B A A t t t t t t t t t t N o o o o o o o o o o o o ω ω ω ω ω ω ω ω ω ω ω ω sin cos sin sin cos sin sin cos cos cos sin cos 1 1 2 2 This is the normal equation of the problem. These equations can be employed to solve for the unknown coefficients. Average Values...
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chapt3 - ME 4905 Advanced Numerical Methods Fourier...

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