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Intro
Vibration and sound are two wave physical phenomena, that is the wave propagation in an elastic and inertial medium.
Vibration -> wave propagation in a solid medium
Sound -> wave propagation in a fluid
To mitigate noise or vibration we can act at 3 levels:
source -> emission 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 string -> string's particles oscillate in a direction that is perpendicular to the wave's propagation direction
- Shear force and bending moment are neglected
- Homogeneous material and constant chain tension
- Tension T is high with respect the string's weight, it means the string's configuration can be approximated as linear 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 fine balance applied on an infinitesimal string element
Fin = max∂2ω/∂t2
m = β A [kg/m]
{
Fin α ≃ α = εβα = ∂ω/∂x
Finx ε2 d2x εβα = ∂/∂x (w + ∂ω/∂xα)
-Fin - Tinx α + Tinner = ϕ
-mα∂2ω/∂t2 - T ∂ω/∂x/,_d/∂x(w + ∂ω/∂x) = ϕ
-m∂2ω/∂t2 + T∂2ω/∂x2 αk = ϕ
=> T∂2ω/∂x2 = m∂2ω/∂t2
X/F RECEIPRANCE
F/X DYNAMIC STIFFNER
X/F MOBILITY
F/X IMPEDANCE
X/F INERTANCE
F/X APPARENT MASS
POINT OF TRANSFER
if the input and output positions are coincident or not
w = kC c = √t/m
- Vo(sin(t) = JωW(ω,t) = Fo eJωt
= so it is in phase with the force
- Vectitive relative
we can compute the driving-point mechanical impedance
zm = F(t) = C= w/fve(t)
= so rinj's characteristic impedance
- Gv it does not depend that is real, constant and freqency impedance at become it is
- independent. it relates force and trend mechanical (100tenp)
- lection @x = x0 in sw
Generally zm is complex and frequency dependent
zm = |zz| + jJi
= Jzm| (cos J + jsinJ) = Jzm|ebJ = zmw)
- |zr| -o relitive breakfast
- |zr| o relutive component
Gv J is here phune diffhonoe between F and V, that
can be remedonted in
the canian's ploes of 2 kztxtive ventos
F
F, = Fo eexjΘ
v = Vo ejxg - Θ
=o
f(t) = ex (f τ ejωt)
= Fo cos (ωt+Θ)
(Vo(t) = ev (VeJωt)) = VoeJωt+Θ-Φ
fin
N + dN/dx dx = N(x + dx, t)
=>
- dN/dx - Md2u/dt2 = 0
N = EA du/dx
=>
EA d2u/dx2 = m d2u/dt2
{ fo + EA (-JK A1 + JK A2) = ∅
EA (-JK A1 eJKL + JK A2 eJKL) + tc JW (A1 e-JKL + A2 eJKL) = ∅
= D A1 = - J Fo / CMN wtc + cmw / wtc (eJ2KL + 1) + cmw (eJ2KL - 1)
= D A1 = J Fo / CMN wtc - cmw / wtc (eJ2KL + 1) + (cmw (eJ2KL - 1)
dynamic eq equation
T + dT/dx + M d2w/dt2 = 0
dT/dx = -M d2w/dt2
=d2M/dx2
deflexion line ep
M/EJ = d2w/dx2
remembering φ = dw/dx
d2w/dx2 = -M/EJ d2w/dt2
Plate
Kirchhoff's Theory
- Flat plate with constant thickness
- The plate is thin, thickness much smaller than the other dimensions
- Homogeneous isotropic and linear elastic material
- No transverse displacement w
- Linear deformation reflected
- Surface where w is supported
- Displacements u and v are proportional to the distance z from the mid plane, because a line segment initially normal to the undeformed xy mid plane remains straight and normal to the elastic surface under bending.
We have to find the bending variation ep of a plate and to do that we first define deformations and moment from an infinitesimal plate element:
Given an infinitesimal rotation dφy due to bending (it is the relative arc between the two small segments), we find the infinitesimal displacement along x axis du.
The same can be done along y for dv.
Obtaining :
=> du = -z dφy
=> dv = -z dφx
kBw = ps⁄B w2 → w = B⁄ps ks2
CB = w⁄KB = 4⁄√psBw = B⁄ps ks
φ = dω⁄dk = 2 B⁄ps KB { φ̇ = 2CB
Plate Eq in Polar Coordinates
r² = x² + y²θ = arctg (y/x)
dx/dx = 1dx/dy = 0
dy/dx = 0dy/dy = 1
dx/dθ = -y/r ²(dy/dθ)(x/y)² = y/x² + (y/x)² = -y/(2) = -hinθ
dy/dθ = x/1+(x/y)²
w((x,u),θ(x,u),t) variable
dx/dt = dw/dt - dx/dθ = cosθ dw/dy - sinθ dw/dx
d²w/dx²
(cosθ dy/dx - sinθ dθ/dx)(cosθ dw/dx - sinθ dw/dy)
= cos²θ d²w/dx - cosθsinθ dw/dy - cosθsinθ d²w/dy + sin²θ d²w/x2
= cos²θ d²w/dx + 2sinθ cosθ dw/dx + 2sinθcosθ
dy/dy = dw/dt - dx/dθ = sinθ dw/dy + cosθ dw/dx
d²w/dx² + d²w/dy² + l d²w/r + 1 d²w/t
Note: 20p w planar coincides to polar coordinates
out Δ= 0 to get a non-trivial sol (imposed motion)
=> Jn(kR) α J'n - In(kR) α J'n = 0
that is a non-linear algebraic eq with unknown k
we can use Bessle’s function properties
α d Jn d In
d In = -k In(kR) + kR I'n(kR)
I Jn(kR)
α d Jn d In
d In = n Jn(kR) - kR J'n(kR)
=> Jn(kR)I'n+In(kR)Jn+Jn(kR)=
algebraic
This non-linear eq is solved for each m value, finding a
consider infinite number of m solutions (kR)nm
=> ωnm = (kR)nm β
------------- √B
The modes unm corresponding to a precise ωnm can be
found from the Bim eq imposing (for example) |C| = 1 and
finding C = - Jn(kR)nm
In(kR)nm
=> Φnm(r,θ) = [ Jn (Krnm . r)
------------------- - Jn(KR)nm In (kR)nm . I)
W(r) ------------------------------
I(n)(kR) I(n)(kR)
{ n gives the number of nodal diameters m gives the number of nodal circumferences
N = 2
M = 1
NB an infinite plate with a point harmonic force applied has a constant load and frequency-independent mechanical impedance
tm = 8βps
it is a good approximation to a finite dimensionNB rec R [[?]] in bending and rot(?)
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