صدا و ارتعاش در صنعت

صدا و ارتعاش در صنعت

اسلاید 1: صدا و ارتعاش در صنعت جلسه دوممحمد رضا منظمmmonazzam@hotmail.com

اسلاید 2: مروري بر رياضيات

اسلاید 3: Real Numbers

اسلاید 4: Indices

اسلاید 5: Logarithms

اسلاید 6: The Binomial Theorem

اسلاید 7: Trigonometry

اسلاید 8: Trigonometry (continue..)

اسلاید 9: Radian Measure

اسلاید 10: Complex Numberscomplex number has both real and imaginary componentMathematical operation is similar to real number e.g.Conjugate of is

اسلاید 11: Modulus (magnitude) and Argument (angle)

اسلاید 12: Polar Form of a Complex NumberThe argument can be written:Therefore

اسلاید 13: Polar Form of a Complex NumberReplacing(Euler’s formula)

اسلاید 14: Polar Form of a Complex Number (continue)Z can be written as:

اسلاید 15: Fourier Analysis (Joseph Fourier (1706-1790)) * Fourier seriesIt enables periodic functions to be represented by infinite series of sine and cosine terms.for a functionFourier series for the function is:

اسلاید 16: Fourier Analysis (Joseph Fourier (1706-1790)) Infinite Fourier Transform and inverse Fourier Transform

اسلاید 17: Fundamental conceptWave: Any moving form-some shape or pattern that travels along without carrying all the medium with it.Some type of wave:Water waveWave on string- musical instrumentMexican waveWind causing wave Heat waveElectromagnetic waveSound wave….

اسلاید 18: Velocity, Frequency and wave lengthThe 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)

اسلاید 19: How Waves TravelA 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

اسلاید 20: Mathematical Description of Harmonic WaveThe disturbance at x1 at time t1 is due to the disturbance at position x0 which occurred at time t0.

اسلاید 21: Mathematical (continue)For harmonic waves ( plane wave):ensures the wave repeats every wavelength

اسلاید 22: Mathematical (continue)Complete representation of a plane wave:To allow the wave to have any value: If we put:

اسلاید 23: Mathematical (continue)Alternative equation for plane wave:

اسلاید 24: Mathematical (continue)Complex Representation

اسلاید 25: Examples (1)If the displacement of the particles of the medium is described by:What is the amplitude, frequency ,wavelength and wave number and what is the speed of the wave?

اسلاید 26: Examples (2)The pressure fluctuations in air are described by:What is the amplitude, frequency, wavelength and the speed of the wave.

اسلاید 27: Examples (3)The pressure, p, is described by: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.

اسلاید 28: Thanks for Listening