Appunti Corso Noise and Vibration Engineering

Name: Appunti Corso Noise and Vibration Engineering
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Appunti completi del corso Noise and Vibration Engineering del prof. Corradi del Politecnico di Milano. Posso fornire anche i laboratori e relativi codici MATLAB. Appunti basati su appunti …

Esame Noise and Vibration Engineering

Facoltà Ingegneria industriale

Dal corso del Prof. Corradi Roberto

Università Politecnico di Milano

A.A. 2020-2021

119 pagine

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INTRO

VIBRATION and SOUND are the wave physical phenomenon, that is the wave propagation in an elastic and inertial medium

VIBRATION - wave propagation is a void medium
SOUND - wave propagation is a fluid

waves transport energy, not mass

To anticipate wave as vibration we can act at 3 levels:

SOURCE - excitation path - RECEIVER

depending on it we have air-borne noise and structure-borne noise, depending on how the energy is transmitted (fluid or solid)

STRING

Transverse wave in a stretched wire - transverse particles oscillate in separate as differentiate that is perpendicular to the wave's propagation direction

HYP:

shear force and bending moment are neglected
homogeneous material and constant cross section
tension T is high with respect the string's weight, it means the string's configuration can be approximated as it is in the static equilibrium position
small vibration amplitude, it means tension's variation are negligible with respect the static configuration value of T
damping is neglected

INTRO

VIBRATION and SOUND are the wave physical phenomena, that is the wave propagation in an elastic and inertial medium

VIBRATION => wave propagation is a void medium

SOUND => wave propagation is a fluid

waves transport energy, not mass

To anticipate rain as vibration we can act at 3 levels:

SOUND => transmission path => receiver

depending on it we have air-borne noise and structure-borne noise, depending on how the energy is transmitted (fluid or solid)

STRING

Transverse wave in a stretched rope => rings particles oscillate in

sections

perpendicular to the wave's propagation direction

shear force and bending moment are neglected
homogeneous material and constant cross section
tension T is high with respect the string's weight, it means the string's configuration can be approximated as identical in the static equilibrium position
small vibration amplitude, it means tension's variations are negligible with respect the static configuration value of T
damping is neglected

The one dimensional wave eq can from a free balance applied on an infinitesimal strings element

Fin = mαx ∂²w/∂t²

m = A [ ^kp/_m/ ]

{

fin α ≅ α = α = ∂w/∂x

fin αx₂ ≅ αx₂ = α_x = ∂/∂x (w + ∂w/∂x αx)

- Fin - T αx + T ∫ inder = 0

- mα ∂²w/∂x² + T ∂²w/∂x² = m ∂²w/∂t²

II order homogeneous PDE

ONE DIMENSIONAL WAVE EQ or VIBRATING TRANE EQ

T d²w / dx² = m d²w / dt²

T W'' = m W.. = => W.. = 1/c² W''

c = √(T/m)

wave propagation speed c(T,m)

standing wave sol

W(x,t) = (x)H(t) => represents synchronous motion , that means string's type of vibration remains the same (x) and it is modulated in time by H(t), the amplitude changes

=> W (x,t) = [A sin (kx) + B cos (kx)] cos (ωt + φ)

it can be the string's free undamped vibration (ω is the natural frequency, free vibrations) or the string is forced undamped vibration (ω is the driving frequency, harmonic input force)

W'' = W.. / c²

=> '' = H'' 1/c²

=> c² '' / = H.. / H = -ω²

=> c²'' + ω² = o => (x) = A sin (kx) + B cos (kx)

=> H.. + ω² H = o => H(t) = D cos (ωt + φ)

ω = kc

W(x,t) = H
wave propagation sol

2 wave, with initially the compact waveform, one progressive and the other regressive. The particular expression of W and W(2) depend on the BCs

W(x,t) = C₁ (ct-x) + C₂ (ct+x)

progressive regressive

=> depending on the problem we can have or only one of

Φ(x) = A sin(kx) + B cos(kx)

ω = kc => k = _ω/^c

Φ'(x) = Ak cos(ux) - Bk sin(ux)

fixed-fixed inp

W(t,Φ) = W(t,ψ) = ∅ => Φ(Φ) = Φ(ι) = ∅

=> B = ∅

=> sin uL(ι) = ∅ = 0

=> KL = nπ

=> W_n L = nπ

=> W_n = ^nπ/L • c^c

=> Φ_n = A sin(^ω_n/_c x)

= A sin(^π/_Lx)

free-fixed inp

W'(t,Φ) = ∅ => A k = ∅ => A = ∅

W(t,L) = ∅ => B cos(uL) = ∅ => KL = (n - ¹/₂) π

= > |W_n| = (n - ¹/₂) π &bul

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