Review of Fourier Series and Its Applications in Mechanical Engineering Analysis


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1 ME 3 Applied Egieerig Aalysis Chapter 6 Review of Fourier Series ad Its Applicatios i Mechaical Egieerig Aalysis TaiRa Hsu, Professor Departmet of Mechaical ad Aerospace Egieerig Sa Jose State Uiversity Sa Jose, Califoria, USA
2 Chapter Outlie Itroductio Mathematical Expressios of Fourier Series Applicatio i egieerig aalysis Covergece of Fourier Series
3 Itroductio Jea Baptiste Joseph Fourier A Frech mathematicia Major cotributios to egieerig aalysis: Mathematical theory of heat coductio (Fourier law of heat coductio i Chapter 3) Fourier series represetig periodical fuctios Fourier trasform Similar to aplace trasform, but for trasformig variables i the rage of ( ad + )  a powerful tool i solvig differetial equatios
4 Periodic Physical Pheomea: Motios of poies Forces o the eedle
5 Machies with Periodic Physical Pheomea A stampig machie ivolvig cyclic puchig of sheet metals Sheet metal Elastic foudatio Mass, M x(t) I a 4stroke iteral combustio egie: Cyclic gas pressures o cyliders, ad forces o coectig rod ad crak shaft
6 Mathematical expressios for periodical sigals from a oscilloscope by Fourier series:
7 The periodic variatio of gas pressure i a 4stoke iteral combustio egie: The PV Diagram P = gas pressure i cyliders Pressure, P Volume, V or Stoke, 5 Itake  Compressio 3 Combustio 4  Expasio 5  Exhaust But the stroke l varies with time of the rotatig crak shaft, so the timevaryig gas pressure is illustrated as: Pressure, P(t) Period T Oe revolutio Period T Next revolutio So, P(t) is a periodic fuctio with period T Time, t
8 FOURIER SERIES The mathematical represetatio of periodic physical pheomea Mathematical expressio for periodic fuctios: If f(x) is a periodic fuctio with variable x i ONE period The f(x) = f(x±) = f(x±4) = f(x± 6) = f(x±8)=.=f(x±) where = ay iteger umber Period: ( , ) or (, ) f(x) x (a) Periodic fuctio with period (, ) Period = : f(t) 3 t t t 3 t (b) Periodic fuctio with period (, )
9 Mathematical Expressios of Fourier Series Required coditios for Fourier series: The mathematical expressio of the periodic fuctio f(x) i oe period must be available The fuctio i oe period is defied i a iterval (c < x < c+) i which c = or ay arbitrarily chose value of x, ad = half period The fuctio f(x) ad its first order derivative f (x) are either cotiuous or piecewise cotiuous i c < x < c+ The mathematical expressio of Fourier series for periodic fuctio f(x) is: f ( x) a = + = a Cos x + b Si x = f ( x ± ) = f ( x ± 4) =... (6.) where a o, a ad b are Fourier coefficiets, to be determied by the followig itegrals: a b c = + x Cos dx c =,,,3,... c = + x Si dx c =,,3,... (6.a) (6.b)
10 Example 6. Derive a Fourier series for a periodic fuctio with period (, ): We realize that the period of this fuctio = () = The half period is = If we choose c = , we will have c+ =  + = Thus, by usig Equatios (6.) ad (6.), we will have: ad a b = a = = + = a c+ = + = = x Cos + b = x Si = x Cos dx x x) Si dx c= = c+ = + f c= = Hece, the Fourier series is: a = + ( a Cos ( x) + b Si( x) ) = with a = Cos ( x) dx =,,,3,... b ( = Si( x) dx =,,3,... (6.3) (6.4a) (6.4b) We otice the period (, ) might ot be practical, but it appears to be commo i may applied math textbooks. Here, we treat it as a special case of Fourier series.
11 Example 6. Derive a Fourier series for a periodic fuctio f(x) with a period (l, l) et us choose c = l, ad the period = l (l) = l, ad the half period = l a x x = + b Si = l = l a b + a Cos = c+ = l+ l = = l = = l x Cos dx l c= l = c+ = l+ l Si c= l = Hece the Fourier series of the periodic fuctio f(x) becomes: x l dx with a b = a + = a Cos x x + b Si (6.5) l l l x = Cos dx =,,,3,... l l l l x = Si dx =,,3,... l l l (6.6a) (6.6b)
12 Example 6.3 Derive a Fourier series for a periodic fuctio f(x) with a period (, ). As i the previous examples, we choose c =, ad half period to be. We will have the Fourier series i the followig form: a x x = + a Cos + b Si = = = a c+ = + = = l b c= c+ = + = = c= x Cos dx = x Si = The correspodig Fourier series thus has the followig form: = a + = a Cos dx x x + b Si (6.7) a = dx (6.8a) x a = Cos dx =,,3,... (6.8b) x b = Si dx =,,3,... (6.8c) Periodic fuctios with periods (, ) are more realistic. Equatios (6.7) ad (6.8) are Thus more practical i egieerig aalysis.
13 The class is ecouraged to study Examples 6.4 ad 6.5 Example (Problem 6.4 ad Problem (3) of Fial exam S9) Derive a fuctio describig the positio of the slidig block M i oe period i a slide mechaism as illustrated below. If the crak rotates at a costat velocity of 5 rpm. (a) Illustrate the periodic fuctio i three periods, ad (b) Derive the appropriate Fourier series describig the positio of the slidig block x(t) i which t is the time i miutes ω Rotatioal velocity ω = 5 RPM Slidig Block, M A B X(t) Crak Radius R Dead Ed Dead Ed A B (X = ) (X = R)
14 Solutio: (a) Illustrate the periodic fuctio i three periods: Determie the agular displacemet of the crak: We realize the relatioship: rpm N = ω/(), ad θ = ωt, where ω = agular velocity ad θ = agular displacemet relatig to the positio of the slidig block For N = 5 rpm, we have: θ t = = 5t Based o oe revolutio (θ=) correspods to /5 mi. We thus have θ = t 5 A ω R θ B Deaded A: x = t = Oe revolutio Deaded B: x = R t = /5 mi Positio of the slidig block alog the xdirectio ca be determied by: x = R RCosθ or x(t) = R RCos(t) = R[ Cos(t)] < t < /5 mi x
15 We have ow derived the periodic fuctio describig the istataeous positio of the slidig block as: x(t) = R[ Cos(t)] < t < /5 mi (a) A ω R θ B x Deaded A: x = t = Oe revolutio Deaded B: x = R t = /5 mi Graphical represetatio of fuctio i Equatio (a) ca be produced as: x(t) R R x(t) = R[ Cos(t)] Θ = / 3/ Time, t (mi) Time t = / /5 mi Oe revolutio d period 3 rd period (oe period)
16 (b) Formulatio of Fourier Series: We have the periodic fuctio: x(t) = R[ Cos(t)] with a period: < t < /5 mi If we choose c = ad period = /5, we will have the Fourier series expressed i the followig form by usig Equatios (6.7) ad (6.8): with ao t t x() t = + acos + bsi = = / = / ao = + [ acost + bsit] = a R Si () ( ) Si( + ) 5 = x t Cos tdt = + + We may obtai coefficiet a o from Equatio (c) to be a o = : (b) (c) The other coefficiet b ca be obtaied by: b 5 ( ) () t Sit dt = R( Cost ) = x R = [ Cos( ) ] + R ( + ) Sit dt [ Cos( + ) ] (d)
17 Covergece of Fourier Series We have leared the mathematical represetatio of periodic fuctios by Fourier series I Equatio (6.): a = + = a x Cos + b Si x = f ( x ± ) = f ( x ± 4) =... (6.) This form requires the summatio of INFINITE umber of terms, which is UNREAISTIC. The questio is HOW MANY terms oe eeds to iclude i the summatio i order to reach a accurate represetatio of the periodic fuctio? The followig example will give some idea o the relatioship of the umber of terms i the Fourier series ad the accurate represetatio of the periodic fuctio : Example 6.6 Derive the Fourier series for the followig periodic fuctio: f t ( t) = S it t
18 This fuctio ca be graphically represeted as: f t () t = S it t f(t) We idetified the period to be: =  () =, ad from Equatio (6.3), we have: a = + ( a Cos ( x) + b Si( x) ) (a) = with = + ( ) ( ) = () + Cos a f t Cos t dt Cos t dt Sit Cos t dt = ( ) for ad ( ) ( ) b = f t Si t dt = () Si t dt + Sit Si t dt =,,3,... or Si( ) t Si( + ) t b = = for + For the case =, the two coefficiets become: Si t a = Sit Cost dt = = ad b = = Sit Sit dt o t
19 f t () t = S it t f(t) The Fourier series for the periodic fuctio with the coefficiets become: f ( t) = + Sit + = ( a Cos t + b Si t ) The Fourier series i Equatio (b) ca be expaded ito the followig ifiite series: Sit Cos t Cos 4t Cos 6t Cos 8t (c) f ( t) = et us ow examie what the fuctio would look like by icludig differet umber of terms i expressio (c): f(t) t (b) Case : Iclude oly oe term: f ( x) = f = Graphically it will look like  Observatio: Not eve closely resemble  The Fourier series with oe term does ot coverge to the fuctio! f = t
20 f t ( t) = S it t f(t) Case : Iclude terms i Expressio (b): Si t f () t = f( t) = + Observatio: A Fourier series with terms has show improvemet i represetig the fuctio Case 3: Iclude 3 terms i Expressio (b): f(t) Si t f ( t) = +  () ( Sit Cos t f t = f3 t) = + 3 Observatio: A Fourier series with 3 terms  represet the fuctio much better tha the two previous cases with ad terms. f 3 ( t) Sit Cos t = + 3 f(t) t t t Coclusio: Fourier series coverges better to the periodic fuctio with more terms icluded i the series. Practical cosideratio: It is ot realistic to iclude ifiite umber of terms i the Fourier series for complete covergece. Normally a approach with terms would be sufficietly accurate i represetig most periodic fuctios
21 Covergece of Fourier Series at Discotiuities of Periodic Fuctios Fourier series i Equatios (6.) to (6.3) coverges to periodic fuctios everywhere except at discotiuities of piecewise cotiuous fuctio such as: f(x) f(x) = < = f (x) < x < x = f (x) x < x <x = f 3 (x) x < x < x 4 The periodic fuctio f(x) has discotiuities at: x o, x, x ad x 4 () f (x) () f (x) x x x x 4 3 (3) Period, The Fourier series for this piecewise cotiuous periodic fuctio will NEVER coverge at these discotiuous poits eve with umber of terms f 3 (x) x The Fourier series i Equatios (6.), (6.) ad (6.3) will coverge every where to the fuctio except these discotiuities, at which the series will coverge HAFWAY of the fuctio values at these discotiuities.
22 Covergece of Fourier Series at Discotiuities of Periodic Fuctios f(x) f (x) () f (x) () x x x x 4 3 (3) x Period, f 3 (x) Covergece of Fourier series at HAFWAY poits: f ) = f ( () f x ) = + f x ) = f ( x ) + f x ) = f 3 ( x4 ) = [ f ( x ) f ( )] ( x [ f ( )] ( 3 x ( 4 f () at Poit () at Poit () at Poit (3) same value as Poit ()
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