Page 722 - Automotive Engineering Powertrain Chassis System and Vehicle Body

Interior noise: Assessment and control C HAPTER 21.1

So p ¼ p 0 þ p 0 (G21.1.1)
r ¼ r þ r 0 (G21.1.2)
0
vr TOT vðr TOT uÞ dxS (F21.1.2) T ¼ T 0 þ T 0 (G21.1.3)
vt dxS ¼ vx
vr TOT vðr TOT uÞ ¼ 0 (F21.1.3) The mass of air between the two plates is given by (r 0 þ
0
vt þ vx r )dx (Morse and Ingard, 1968). The net force acting on
this mass is p(x) p(x þ dx). Using Newton’s second law,
This equation of mass conservation may be linearised
thus: this must be equal to the mass times the acceleration of
the fluid.
r TOT u ¼ðr þ rÞu (F21.1.4)
0
r TOT u ¼ r u þ ru pðxÞ pðx þ dxÞ¼ vp dx ¼ du ðr þ r Þdx
0
0
vx dt 0
The ru term is the product of two small quantities and (G21.1.4)
one might choose to neglect it and so
Now the total differential du may be expressed in
dx
r TOT u z r u (F21.1.5) its partial differential form where u ¼ f(x, t)and the
0
total change in u is given by the sum of the partial
As r 0 is not a function of x:
changes:
vðr TOT uÞ vu
vx z r 0 vx (F21.1.6) vu vu
du ¼ dx þ dtðWeltner et al: ð1986Þ for exampleÞ
As r 0 is not a function of t either, so vx vt
vr TOT vr (F21.1.7)
vt ¼ vt So, dividing both sides by dt in the limit dt/ 0

and the linearised mass conservation equation becomes
du vu dx vu dt
¼ $ þ $
vr vu dt vx dt vt dt
þ r 0 ¼ 0 (F21.1.8)
vt vx du ¼ u vu þ vu
dt vx vt
Re-arranging
So, from equation (G21.1.4)
vu 1 vr
¼ (F21.1.9)
vx r vt vp vu vu
0
0
dx ¼ u þ ðr þ r Þdx
0
1 1 vp vx vx vt
Now ¼ from (G21.1.5)
r 0 k 0 vr vp ¼ðr þ r Þ vu þ vu
0
(F21.1.10) vx 0 vt vx
vp vu 1 vp vr
k ¼ r 0 ¼ This is the non-linear inviscid Euler equation (Kinsler
0
vr vx k 0 vr vt
et al., 1982).
r 0
So finally the linerised mass conservation equation is: vu vu r 0
Now if u vx vt and the condensation ðs ¼ r 0 Þ 1,
vu 1 vp the non-linear inviscid Euler equation reduces to its
¼ (F21.1.11)
vx k 0 vt linearised form, being
vp vu
¼ r 0 (G21.1.6)
Appendix 21.1G: The derivation of vx vt
the non-linear (and linearised) From this
inviscid Euler equation 1 ð vp
uðx; tÞ¼ dt (G21.1.7)
r 0 vx
Take two plane surfaces – one at x and the other at
x þ dx, each one with unit area. There is an acoustic wave So, a relationship between pressure gradient and
causing: particle velocity is found.

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