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