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CSUN ME 501B - Fourier Series

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Fourier Series January 26, 2009ME 501B – Engineering Analysis 1Fourier SeriesFourier SeriesLarry CarettoMechanical Engineering 501BSeminar in Engineering AnalysisJanuary 26, 20092Outline• Review last class• Fourier series as expansions in periodic functions– Comparison to eigenfunction expansions• Odd and even functions• Periodic extensions of non-periodic functions• Complex Fourier series3Review Last Lecture• Discussed Sturm-Liouville Problem• Solutions are a set of orthogonal eigenfunctions, ym(x) that can be used to express other functions, f(x)∑∞==0)()(mmmxyaxf()∫∫==bammbammmmmdxxyxyxpdxxfxyxpyyfya)()()()()()(,),(• p(x) is weight function• Have to compute am4Review Orthogonal Functions• Defined in terms of inner product• Norm of two like eigenfunctions ||yi||()ijibajijiydxxpxyxyyyδ2*)()()(, ==∫()ijbajijidxxpxfxfffδ==∫)()()(,*• Convert orthogonal eigenfunctions to orthonormal eigenfunctions• Orthonormal eigenfunctionsiiiyyf =5Review Sturm-LiouvilleGeneral equation whose solutions provide orthogonal eigenfunctions– Defined for a ≤ x ≤ b with p(x), q(x) and r(x) continuous and p(x) > 0– (Homogenous) differential equation and boundary conditions shown below[]0)()()(=++⎟⎠⎞⎜⎝⎛yxpxqdxdyxrdxdλ0)(0)(2121=+=+==bxaxdxdybydxdykaykll6Review Sturm-Liouville Results• Eigenvalues are real• Eigenfunctions defined over a region a ≤x ≤ b form an orthogonal set over that region.• Eigenfunctions form a complete set over an infinite-dimensional vector space• We can expand any function over the region in which the Sturm-Liouville problem is defined in terms of the eigenfunctions for that problemFourier Series January 26, 2009ME 501B – Engineering Analysis 27Review Eigenfunction Expansions• Eigenfunction expansion formula∑∞==0)()(mmmxyaxf()∫∫==bammbammmmmdxxyxyxpdxxfxyxpyyfya)()()()()()(,),(• Equation for amcoefficients in eigenfunction expansion of f(x)8Review Expansion of f(x) = x• Start with general equation for am• Use ym= sin(mπx) over 0 ≤ x ≤ 1 which is a Sturm-Liouville solution• Weight function p(x) = 1()π−=π=ππ==+∫∫∫mdxxmxdxxmdxxmxyyfyammmmm11010210)1(221)sin()(sin)sin(,),(⎥⎦⎤⎢⎣⎡+−+−== L4)4sin(3)3sin(2)2sin(1)sin(2)(xxxxxxfπππππ9Review Partial Sums – Small00.10.20.30.40.50.60.70.80.911.10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xSeries sumEx a ct1 term2 terms3 terms5 terms10 terms10Review Partial Sums – Large00.10.20.30.40.50.60.70.80.911.11.20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xSeries sumsExact10 terms25 terms50 terms100 terms11Fourier Series• Have same basic idea as eigenfunction expansions– Represent other functions, f(x), as a series of sines and cosines– Compute coefficients in a similar way to eigenfunction expansions– Fourier series based on periodicity of trigonometric functions12Fourier Series• Equations for series and coefficients defined for –L < x < L∑∞=⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+=10sincos)(nnnLxnbLxnaaxfππ∫−=LLdxxfLa )(210∫−⎟⎠⎞⎜⎝⎛=LLndxLxnxfLbπsin)(1∫−⎟⎠⎞⎜⎝⎛=LLndxLxnxfLaπcos)(1Fourier Series January 26, 2009ME 501B – Engineering Analysis 313Basis for Fourier Series• Applies to periodic functions• Must be piecewise continuous• Derivative must exist at all points in the period• At discontinuity both left-hand and right-hand derivatives exist• Fourier series converges to f(x)• At discontinuity the series converges to average of f(x-) and f(x+)14Even and Odd Functions• Odd function: f(–x) = – f(x) (like sine)• Even function g(–x) = g(x) (like cosine)sinecosine• For odd f(x)0)( =∫−LLdxxf• For even g(x)∫∫=−LLLdxxgdxxg0)(2)(15Even and Odd Functions II• The product of an even function, g(x) = g(–x) and an odd function f(–x) = –f(x) is an odd function• Proof: define h(x) = f(x) · g(x)• h(–x) = f(–x) · g(–x) = –f(x) · g(x) = –h(x) • Since sine is an odd function, bn= 0 if f(x) is even, i.e., f(–x) = f(–x)∫−⎟⎠⎞⎜⎝⎛=LLndxLxnxfLbπsin)(1• Fourier coefficient16Even and Odd Functions III• For odd functions, the Fourier series coefficients a0and anare zero • Can have half-range expansions that consider only 0 < x < L• Apply to non periodic functions to get “periodic extensions”∫−=LLdxxfLa )(210∫−⎟⎠⎞⎜⎝⎛=LLndxLxnxfLaπcos)(117Even and Odd Fourier Series• We can use relations for even and odd functions to transform Fourier coefficient equations• For odd functions, f(–x) = –f(x) we have only sine terms with coefficients bn• Even functions have only cosine (and constant) terms with coefficients a0and an• Next chart shows series and coefficients 18Odd Function Sine Series• f(x) is an odd function f(–x) = –f(x)• Sine only series defined for –L ≤ x ≤ L•bncomputed as integral from 0 to L∑∞=⎟⎠⎞⎜⎝⎛=1sin)(nnLxnbxfπ∫∫⎟⎠⎞⎜⎝⎛π=⎟⎠⎞⎜⎝⎛π=−LLLndxLxnxfLdxLxnxfLb0sin)(2sin)(1Fourier Series January 26, 2009ME 501B – Engineering Analysis 419Even Function Cosine Series• f(x) is an even function f(–x) = f(x)• Cosine only series defined for –L ≤ x ≤ L•anevaluated by integral from 0 to L∑∞=⎟⎠⎞⎜⎝⎛+=10cos)(nnLxnaaxfπ∫∫==−LLLdxxfLdxxfLa00)(1)(21∫∫⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛=−LLLndxLxnxfLdxLxnxfLa0cos)(2cos)(1ππ20Half-interval series• Based on equations for sine series for odd functions and cosine series for even functions• Either the sine or cosine series can apply to any function– Can use sines to expand even functions– Can use cosines to expand odd functions• Each series defined for 0 ≤ x ≤ L• Behavior outside region 0 ≤ x ≤ L depends on the function21Half-interval Series Example• Last class computed series for f(x) = x in terms of sin(nπx/L) with L = 1 for 0 ≤ x ≤ L• Would get same coefficients from equations for Fourier sine series• Get correct result for –L ≤ x ≤ L with periodic extensions∑∞=⎟⎠⎞⎜⎝⎛=1sin)(nnLxnbxfπ()π−=⎟⎠⎞⎜⎝⎛π=⎟⎠⎞⎜⎝⎛π=+∫∫nLdxLxnxLdxLxnxfLbnLLn10012sin2sin)(2Fourier Sine Series for f(x) = x-1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.50.60.70.80.91.01.1-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0xf(x)Exact1 term2 terms5 terms10 terms23Cosine Series for f(x) = x• Set f(x) = x in equations for a0and


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