صدا و ارتعاش در صنعت

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صدا و ارتعاش در صنعت جلسه دوم محمد رضا منظم ‎mmonazzam@hotmail.com‏ 

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eo مروري بر ریاضیات 

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2 Real Numbers at(btd=(atD+c , 240-6 atb=b+a_ , ab=ba db+d=abtac , ‏اك‎ bay Ava sa 

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mt 1 ea 4 Indices (at) ‏تم‎ a =a Faden =a aes 

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Logarithms yo=x ee n=log, vy log| xy} =log,x+log,y , log, 3 ‏ب 7ر10 1 رومت‎ log, x"| =mdog, x el _ log, x log, x= 7 log, * Tog, n log, n 

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53 The Binomial Theorem (a+ D? =a? +2ab+ b (a+ D® =a? + 3a’b+ 3ab +B ‎ey ad‏ جات ویر "و +ج) ‎dn- ( 9 2‏ ‎)1+ ‏بر ج... + لیر جرور +1- "لر‎ n>0 ‎ ‎dn- 1(...)0- ‏و(‎ ‎2 ‎(14+ 9” =1+ nx+...+ all n 

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2 Trigonometry 0 6010 - 0 , cosed = 2 ‏ار‎ 9008 1 cod tam sing 003 ‎6)=-sin, co$- 6) =co#, tat 6) =- tam‏ نزو ‎sir’ 9+cos @ =1‏ ‎1+ tari ‏م6‎ =secéa cot @ +1=coseca ‎ 

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Trigonometry (continue..) it A+B =sinAcosB+cosdsinB co$A+B =cosAcosBFsinAsinB 00 ‏ا‎ ‎1+tanAtanB ‎1 4 ۰ 2tanA co@A=cos A- sit A sin2A=2sinAcos4 tarA=. 1- tart A J 4 1 2 3 ۰ 78 = 3 ۳ X+ 23 SURE SIY ‏بای‎ sinx- SHY ‏رت‎ cosx+ cosy=2cos=-* cos”, 005 005۲ - ‏مور‎ sinX ‏ال‎ 

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Radian Measure 360 =27 raa 240 =* rad cos? =0, > @ =nt ifsim) =sina= 0 ‏وع‎ +) ۵ if cos) =cosa> 0 =2m +a iftam=tam = @=m+a a x x=sim) = 0=sin'x ‏ع وت‎ 2 2 x=cos = @ =cos'x 0-6 >> xstaw = 0 =tan'x pee gre 2 2 

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Complex Numbers ¥64=8-8 y= 64=8/-1 1 04-9 fol pai ‏از‎ <1 ‏اه رم لت لح سا ها موه ویو‎ Z=a+ jb Oukexvuncd operatic ty ‏موی او با نوا‎ ec. ie cone (a+ jBlc+ jd =(ac- bd + fad+ bd 22 ‏نه لال اك‎ + at jb_(a+ jhlc- jd _ ‏و‎ ‎c+ jd (c+ jdlc- jd e+ ne + Abe ad لك ا 0ك )2 2125 ) 

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Modulus (magnitude) and Argument ‏و‎ + 22 ‎ol‏ ی ره @=arg=tan'2 - > *>7 

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4 Polar Form of a Complex Number = The argument can be written: 6098 ay sind -> - ۲ > > 4 3 = Therefore z=|Acos) + jsind) ZZ, =|z\|z,| coo, +0,)+ jsio, +0,)| A ‏ا‎ - 6) + jsila, - ‏آ(یه‎ 2 |Z 

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Polar Form of a Complex 2 Ce et ata 2! cosx=1- = tea 13 oe we a a! ۳ 9 eel (red! (cers Pore) بت رح زو ‎OSB‏ R xbyjo in thee* 2 pf ‏تس ۳ -1|- ام‎ 2 4 Boor AS 0- ‏م‎ 7 ‏ا‎ =cos# + jsind aa 

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Polar Form of a Complex Number (continue) = Zcan be written as: ae 0 ‏لور دج‎ مصزوز + ومع و ‎jsind‏ -وووع و مرج - 6 ‎sing SOs eae 2 ‎e+e” ‎cos? = 2 

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Fourier Analysis (Joseph Fourier (1706-1790) * Fourier series It enables periodic functions to be represented by infinite series of sine and cosine terms. 8 ‏07م مع كر د‎ 5 for a function (Powter series Por te Rauctoa ‏ضجا‎ ‎fo gat dle cosit+ b, sinnot| ‏عا‎ 2 an ‎a Dcosnwtdt‏ اج رت ‎rs‏ ‎2 ‏فا‎ ‏رك‎ - sf sinmotdt 

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Fourier Analysis (Joseph Fourier (1706-1790)) = Infinite Fourier Transform and inverse Fourier Transform gsi] =Xo) = fle at ecll 1 5 ‏اس‎ ‎é | X(a)] i dt 

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Fundamental concept * Any moving form-some shape or pattern that travels along without carrying all the medium with it. Some type of wave: Water wave Wave on string- musical instrument Mexican wave Wind causing wave Heat wave Electromagnetic wave Sound wave 

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Velocity, Frequency and wave length ايا ومسي افا i /۱ ۷ {\ re 5 a Hoque Hoe i Te nea = The velocity of a wave (c): the speed at which its wave-form travels along, the speed of any labelled part of the disturbance. " The frequency of a wave (f): The number of oscillation it makes in 1 second. In 1 second the wave has travelled "c” metres so that ‏"ع"‎ metres contains “f” cycles of the wave. Hence in space one complete wave is ¢/f metres long (wavelength A) The time of one oscillation is the period (T) c a tod, 0 6 

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How Waves Travel =" A wave travels essentially because: = One piece of the medium disturbed by the wave disturbs the next piece of medium ahead and gives up the motion to it. = The waves are: = Longitudinal: The pieces of the medium oscillate in the same direction as the wave propagate. = Transverse: The pieces of the medium oscillate perpendicular to the direction propagation of the wave. = Some other types including, Shear and Bending 

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Mathematical Description of ۳ Harmonic Wave = The disturbance at x: at time ti is due to the disturbance at position xo which occurred at time to (% - x) 45 ‏و‎ 200080 وا - 0 < ود 01 

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4 Mathematical (continue) = For harmonic waves ( plane wave): ‏انم بر‎ Flot 0 2051117611132 S25 Axil 3 ct+ x) ۱ negativdirectic 20 xwures the wave repes every waveleonth 

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4 Mathematical (continue) = Complete representation of a plane wave: To allow the wave to have any value: y= Axil (ct | + Boot (ct 3] ‘Pwe mt A=Dcosy) and — B=Dsirlg) y=Dcoky) xsi (ct 5 + ‏أده (ولشوط‎ (ct a , where D=V A’ +B andtap) =3 y=Dsif (ct ‏ور‎ + 

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۳ Mathematical (continue) = Alternative equation for plane wave: : ۶-9 and w=2f ‎ot- Fxeg‏ فده دير ‎or‏ ‎y=Cxsirlot- kx 9), k 00 (waveumber 

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۳ Mathematical (continue) = Complex Representation ae ‏إلدع ار‎ ‏یزرو‎ A=A+jA or y=(4.+ JA) co} 2 (ct x + joi "Ect 01 and the real par. ya Aco} 2 (op | Asi 2 (ct- a| 

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Examples (1) = If the displacement of the particles of the medium is described by: 0-0005 207 + = What is the amplitude, frequency ,wavelength and wave number and what is the speed of the wave? 

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+ Examples (2) = The pressure fluctuations in air are described by: p=0.01co$20G@t- 1.85x) + 0.005sir 20Gt- 59 = What is the amplitude, frequency, wavelength and the speed of the wave. 

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4 Examples (3) = The pressure, p, is described by: p= ‏+مالويم‎ kt = |f the pressure amplitude is 0.01 pa and at t=0 , X=0 the value of p is 0.005 pa find Ar and Ai. 

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{oe Thanks for Listening 

صدا و ارتعاش در صنعت جلسه دوم محمد رضا منظم ‏mmonazzam@hotmail.com مروري بر رياضيات Real Numbers a  b c a  b c , a  b b  a ab  c ab ac , , abc abc abba b c b c   a a a a 0 Indices a  a m n m n mn , a  n am a 1 0 , am m n , a n a , am.an amn a n  1 a n Logarithms n y x logn xy logn x  logn y n logx y  ,  x logn   logn x  logn y  y   logn xm mlogn x 1 logn x  logx n , logm x logn x  logm n , The Binomial Theorem a  b 2 a  b 3 2 2 a  2ab b 3 2 2 3 a  3a b  3ab  b nn  1...n  r  1 n r r a  b a  na b  ... a b  .... bn r! n 1 x n n 1 nx n 1 nn  1 2! 2 x  nn  1n  2 3! x3  .... xn nn  1....n  r  1 r 1 x 1 nx ... x  .... r! n n 0 all n Trigonometry sin 1 tan  , cot  , cos tan sin    sin , 1 cosec  , sin cos   cos , tan    tan sin2   cos2  1 1 tan  sec  2 2 cot2   1 cosec2 1 sec  cos Trigonometry (continue..) sinA B sinAcosB cosAsinB, cosA B cosAcosB sinAsinB AB  tanAtanB tan 1tanAtanB cos2A cos2 A  sin2 A, sin2A 2sinAcosA, 2tanA tan2A  1 tan2 A x y x y x y x y sinx  siny 2sin cos , sinx  siny 2cos sin 2 2 2 2 x y x y x y x y cosx  cosy 2cos cos , cosx  cosy  2sin sin 2 2 2 2 Radian Measure 360 2 rad 0 4 240  rad 3 0 cos 0,   n if sin sina   n   1n a if cos cosa   2n a if tan  tana   n  a x sin   sin 1 x     2 2 x cos   cos 1 x 0   x  tan    tan 1 x     2 2 Complex Numbers 64 8, 8 j 2  1  64 8  1 j 3  j  64 8 j j 4 1 complex number has both real and imaginary component z a  jb Mathematical operation is similar to real number e.g. a  jb c  jd a  c  jb  d Conjugate of a  jbc  jd ac bd jad  bc z is z* zz* a  jba  jb a2  b2 a  jb a  jbc  jd 1   2 2 ac bd  jbc ad c  jd c  jdc  jd c  d z1  z2  * * 1 * 2 z  z , z1z2  * * * 1 2 z z , z  z  n * * n Modulus (magnitude) and Argument (angle) z  a  jb  a2  b2 z  z, zz  z * * b  argz  tan a 1 2     Polar Form of a Complex Number  The argument can be written: cos   a , z sin  b z     Therefore z  z cos  j sin  z1z2  z1 z2 cos1   2   j sin1   2  z1 z1 cos1   2   j sin1   2   z2 z2 Polar Form of a Complex Number r x ex 1 x   ...  ... 2! r! x2 2r x r cosx 1   ...  1  .... 2r! 2! 4! x2 x4 x2r1 sinx  x    ...  1  .... 2r  1! 3! 5! x3 Replacing 2 4    e j 1    2! 4! x5 r x by j in the ex  ...    3 5 j     3! 5!  (Euler’s formula)  .. cos  j sin  Polar Form of a Complex Number (continue)  Z can be written as: z  ze j e j cos  j sin e j cos  j sin e j  e sin  2j cos  j e j  e 2 j Fourier Analysis (Joseph Fourier (1706-1790)) *   Fourier series It enables periodic functions to be represented by infinite series of sine and cosine terms. for a function f t  f t  nT Fourier series for the function is:  1 f t  a0   an cosnt  bn sinnt 2 Tn1 a0  an  bn  2 T 2 T 2 T f tdt 0 T f tcosnt dt 0 T f tsinnt dt 0 Fourier Analysis (Joseph Fourier (1706-1790))  Infinite Fourier Transform and inverse Fourier Transform   jt          x t  X   x t e dt  1 1   X    xt  2  jt   X  e dt   Fundamental concept  Wave:   Any moving form-some shape or pattern that travels along without carrying all the medium with it. Some type of wave:         Water wave Wave on string- musical instrument Mexican wave Wind causing wave Heat wave Electromagnetic wave Sound wave …. Velocity, Frequency and wave length   The velocity of a wave (c):  the speed at which its wave-form travels along, the speed of any labelled part of the disturbance. The frequency of a wave (f):  The number of oscillation it makes in 1 second.  In 1 second the wave has travelled “c” metres so that “c” metres contains “f” cycles of the wave.  Hence in space one complete wave is c/f metres long (wavelength λ)  The time of one oscillation is the period (T) c  , f c  f  , T 1 f How Waves Travel  A wave travels essentially because:   One piece of the medium disturbed by the wave disturbs the next piece of medium ahead and gives up the motion to it. The waves are :    Longitudinal: The pieces of the medium oscillate in the same direction as the wave propagate. Transverse: The pieces of the medium oscillate perpendicular to the direction propagation of the wave. Some other types including, Shear and Bending Mathematical Description of Harmonic Wave  The disturbance at x1 at time t1 is due to the disturbance at position x0 which occurred at time t0. t1 t0  x1  x0  c ct1  x1 ct0  x0 constant Mathematical (continue) For harmonic waves ( plane wave):  2  ct x y  Asin postivedirection      2 ct x y  Asin    2  negative direction ensures the wave repeats every wavelength Mathematical (continue)  Complete representation of a plane wave: To allow the wave to have any value:  2   2  y  Asin ct x  Bcos ct x       If we put: A  D cos  and B  D sin   2 ct x  D sin cos 2 ct x       y  D cos sin  2 ct x  , where D  A2  B2 and tan   B A    y  D sin Mathematical (continue)  Alternative equation for plane wave:   y C sin t  2    x   , f  c and or y C sint  kx  , k 2  (wavenumber )  2f Mathematical (continue)  Complex Representation  2 ct x     j  j t kx y  Ae y  Ae or , or  A  Ar  jAi   2 ct x  j sin 2 ct x  y Ar  jAi  cos         2 ct x  Ai sin 2 ct x       y  Ar cos and the real part Examples (1)  If the displacement of the particles of the medium is described by:   d 0.005sin 20t  x   2   What is the amplitude, frequency ,wavelength and wave number and what is the speed of the wave? Examples (2)  The pressure fluctuations in air are described by: p 0.01cos200t  1.85x  0.005sin200t  1.85x  What is the amplitude, frequency, wavelength and the speed of the wave. Examples (3)  The pressure, p, is described by:  p  Ae  j t kx If the pressure amplitude is 0.01 pa and at t=0 , x=0 the value of p is 0.005 pa find Ar and Ai. Thanks for Listening