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 Tai-Ra 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 4-stroke 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 4-stoke iteral combustio egie: The P-V 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 time-varyig 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 piece-wise 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 Dead-ed A: x = t = Oe revolutio Dead-ed B: x = R t = /5 mi Positio of the slidig block alog the x-directio 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 Dead-ed A: x = t = Oe revolutio Dead-ed 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 piece-wise 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 piece-wise 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 HAF-WAY 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 HAF-WAY 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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