Page 720 - Automotive Engineering Powertrain Chassis System and Vehicle Body

Interior noise: Assessment and control C HAPTER 21.1

Now, taking the linearised inviscid Euler equation Substituting equation (E21.1.10) into equation (E21.1.9)
(derived in Appendix 21.1G)

Ae ikL þ x 1 Ae ikL
xþ1
vp vu Z MO
¼ r 0 (E21.1.2) ¼ Y
vx vt r c 0 S ikL x 1 ikL
0
Ae Ae
xþ1
From this:
ð Dividing both numerator and denominator by A
1 vp
uðx; tÞ¼ dt (E21.1.3)
r 0 vx e ikL þ x 1 ikL
e
Y ¼ xþ1 (E21.1.11)
So, a relationship is found between pressure gradient and e ikL x 1 e ikL
particle velocity. xþ1
For harmonic waves the integral with respect to time
is given by Morse and Ingard (1968), 1 . Multiplying both numerator and denominator of
iu equation (E21.1.11) by (xþ 1)
So,
e ikL x þ e ikL þ e ikL x e ikL
1 vp Y ¼ e ikL x þ e ikL e ikL x þ e ikL (E21.1 12)
uðx; tÞ¼ (E21.1.4)
iur vx
0
Now, there are these two standard relationships (see
vp i½utþkðL xފ i½ut kðL xފ
¼ ikðL xÞAe ikðL xÞBe Weltner et al. [1986] for example)
vx
ix
(E21.1.5) e e ix
¼ sin x (E21.1.13)
2i
As k ¼ uc
ix
e þe ix
1 2 ¼ cos x (E21.1.13a)
uðx; tÞ¼ Ae i½utþkðL xފ Be i½ut kðL xފ
r c 0 Substituting equations (E21.1.13) and (E21.1.13a)
0
(E21.1.6) into equation (E21.1.12)
Now
x e ikL þ e ikL þ e ikL e ikL
Force Y ¼
Z ML ¼ (E21.1.7) x e ikL e ikL þ e ikL þ e ikL
u
2x cos kL þ i2 sin kL
Y ¼ (E21.1.14)
2 cos kL þ i2x sin kL
Force ¼ PðL; tÞ: S
Divide both the numerator and the denominator of
A þ B equation (E21.1.14) by 2 cos kL to get
Z ML ¼ r c 0 S (E21.1.8)
0
A B
x þ i tan kL
The input mechanical impedance is given by: Y ¼ 1 þ ix tan kL
Ae ikL þ Be ikL Z ML þ i tan kL
Z MO ¼ r c 0 S (E21.1.9) Z MO ¼ r 0 c 0 S ½Kinsler et al:; 1982Š
0
Ae ikL Be ikL r c 0 S 1 þ i r 0 c 0 S tan kL
Z ML
0
The two equations may be combined to eliminate both (E21.1.15)
A and B from equation (E21.1.8)
Z ML
Now is a complex term
0
A þ B Z ML r c 0 S
x ¼ ¼ x
A B r c 0 S Z ML
0
xA xB ¼ A þ B r c 0 S ¼ r þ ix (E21.1.16)
0
xA A ¼ xB þ B
ðx 1Þ r ¼ acoustic (flow) resistance
B ¼ A (E21.1.10)
x þ 1 x ¼ auoustic (flow) resistance
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